Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

Journal of Functional Analysis 236 (2006) 369–394 

www.elsevier.com/locate/jfa 

Estimation optimale du gradient du semi-groupe 

de la chaleur sur le groupe de Heisenberg 

Hong-Quan Li ∗ 

SFB 611, Institut für Angewandte Mathematik, Universität Bonn, Poppelsdorfer Allee 82, D-53115 Bonn, Germany 

Reçu le 6 juillet 2005 ; accepté le 21 février 2006 

Disponible sur Internet le 18 avril 2006 

Communiqué par G. Pisier 

Résumé 

En utilisant l’inégalité de Poincaré et la formule de représentation, on montre que sur le groupe de Heisenberg 

de dimension réelle 3, H1 , il existe une constante C>0 telle que : 

 

∇e t f (g) Ce t |∇f | (g), ∀g ∈ H 1 ,t>0, f∈ C ∞ o 

H 1 . 

Ce résultat répond par l’affirmation à la question ouverte de [B.K. Driver, T. Melcher, Hypoelliptic heat 

kernel inequalities on the Heisenberg group, J. Funct. Anal. 221 (2005) 340–365]. Aussi, le résultat principal 

de [B.K. Driver, T. Melcher, Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal. 

221 (2005) 340–365] peut être considéré comme une conséquence immédiate de l’inégalité précédente. 

© 2006 Elsevier Inc. All rights reserved. 

Mots-clés : Groupe de Heisenberg ; Semi-groupe de la chaleur ; Inégalité de Poincaré ; Formule de représentation ; 

Noyau de la chaleur ; Structure sous-riemannienne 

1. Introduction 

Sur R n , notons ∇ le gradient, et e t (t>0) le semi-groupe de la chaleur. On voit que ∇ et 

e t commutent, i.e. 

* Nouvelle adresse : Department of Mathematics, Fudan University, 220 Handan Road, Shanghai 200433, People’s 

Republic of China. 

Adresse e-mail : li-hq@wiener.iam.uni-bonn.de. 

0022-1236/$ – see front matter © 2006 Elsevier Inc. All rights reserved. 

doi:10.1016/j.jfa.2006.02.016

370 H.-Q. Li / Journal of Functional Analysis 236 (2006) 369–394 

∇e t f(g)= e t (∇f )(g), ∀g ∈ R n ,t>0, f∈ C ∞ o 

R n . 

En général, cette propriété ne reste plus valable dans le cadre des variétés riemanniennes complètes, 

(M, gM). Cependant, si la courbure de Ricci sur M, Ric(M), est minorée par k (k ∈ R), 

alors D. Bakry a montré dans [4] que : 

 

∇e t f(g) e −kt e t |∇f | (g), ∀g ∈ M, t > 0, f∈ C ∞ o (M). (1.1) 

Lorsque la courbure de Ricci sur M est minorée, M est stochastiquement complète, i.e. 

et1 = 1 pour tout t>0. Une conséquence immédiate de (1.1) est que pour tout 1 w 0, f∈ Co (M), (1.2) 

ou bien, plus faiblement, 

 

∇e t f 

L∞ e −kt ∇f L∞, ∀t >0, f∈ C∞ o (M). (1.3) 

Les estimations de type (1.2) ou bien sous une forme affaiblie (ou plus forte) ont beaucoup 

d’applications, par exemple, dans l’étude des inégalités de Poincaré et de Sobolev logarithmique 

pour le semi-groupe de la chaleur, dans l’étude du critère Γ2 et aussi dans l’étude des transformées 

de Riesz, voir par exemple [1–3,7] ainsi que leurs références. 

Récemment, von Renesse et Sturm ont montré que dans le cadre des variétés riemanniennes 

complètes, l’estimation (1.3) avec k ∈ R implique que Ric(M) k, voir [24]. Pour d’autres 

résultats équivalents à l’estimation (1.1), voir par exemple [1,24] ainsi que leurs références. 

Et dans [17] (voir aussi [18]), on a montré que les estimations (1.3) ou bien (1.2) ne restent 

plus valables sur les variétés coniques. Plus précisement, sur le cône C(Sϑ) avec ϑ>2π, qui est 

une variété riemannienne à courbure sectionnelle nulle mais pas complète, il existe une fonction 

lisse à support compact définie sur C(Sϑ), fo, telle que ∇etfoL∞ =+∞pour tout t>0. 

Dans [9], Driver et Melcher ont étudié les estimations de type (1.2) dans le cadre du groupe 

de Heisenberg de dimension réelle 3, H1 , qui peut être considéré comme une variété munie d’un 

Laplacien dégénéré. 

Rappelons, voir par exemple [12, p. 98], que H1 = R2 × R est un groupe de Lie stratifié pour 

la loi 

(x1,x2,t)· (x ′ 1 ,x′ 2 ,t′ ) = x1 + x ′ 1 ,x2 + x ′ 2 ,t1 + t ′ 1 + 2(x′ 1 x2 − x1x ′ 2 ) . 

Et le sous-Laplacien sur H 1 s’écrit comme 

= X 2 1 + X2 2 , 

où les deux champs de vecteurs invariants à gauche sur H 1 ,X1 et X2, sont définis par 

X1 = ∂ ∂ 

+ 2x2 

∂x1 ∂t , X2 = ∂ ∂ 

− 2x1 

∂x2 ∂t . 

Notons ∇=(X1, X2) le gradient, et e t (t>0) le semi-groupe de la chaleur sur H 1 .

H.-Q. Li / Journal of Functional Analysis 236 (2006) 369–394 371 

Par une méthode probabiliste, Driver et Melcher ont montré que pour tout 1 1) telle que : 

 

∇e t f 

(g) 

t 

Cw e |∇f | w (g) 1/w 1 ∞ 1 

, ∀g ∈ H ,t>0, f∈ Co H . (1.4) 

Ils ont montré aussi que leur méthode ne marche pas pour le cas où w = 1, qui reste ouvert. 

De plus, Melcher a donné dans [23] quelques familles des fonctions sur lesquelles l’estimation 

précédente avec w = 1 est satisfaite. 

On remarque aussi que Lust-Piquard et Villani ont montré dans [21] les estimations (1.4) 

(avec 1 0, f∈ C ∞ o 

H 1 . 

Par le Théorème 1.1, on obtient immédiatement l’inégalité de Sobolev logarithmique pour le 

semi-groupe de la chaleur comme suit : 

Corollaire 1.2. Soit C1 > 1 comme dans le Théorème 1.1. Alors, pour tout t>0 et toute f ∈ 

C ∞ o (H1 ),ona: 

et aussi 

e t f 2 log f 2 (g) − e t f 2 (g) log e t f 2 (g) C 2 1 tet |∇f | 2 (g), ∀g ∈ H 1 , 

e t f 2 log f 2 (g) − e t f 2 (g) log e t f 2 (g) C 2 1t |∇etf 2 | 2 (g) 

etf 2 , ∀g ∈ H 

(g) 

1 . 

Pour montrer ce corollaire, il suffit de modifier un peu la preuve de la partie (i) ⇒ (ii) du 

Théorème 5.4.7 de [1, pp. 85–86]. 

L’idée de la démonstration du Théorème 1.1 est d’utiliser l’inégalité de Poincaré et la formule 

de représentation qu’on rappelle dans la Section 2 (les lecteurs attentifs trouveront qu’en fait, la 

propriété du doublement du volume localement et la formule de représentation localement sont 

suffisantes). Cette idée provient d’une observation dans le cadre des espaces euclidiens et c’est 

très facile à comprendre. Cependant, la démonstration du Théorème 1.1 est relativement difficile : 

on doit utiliser la structure sous-riemannienne de H 1 (i.e. l’expression explicite de la distance de 

Carnot–Carathéodory et de la géodésique) ainsi que les estimations optimales du noyau de la 

chaleur et de son gradient, qu’on rappelle ou donne dans la Section 3. 

1.1. Quelques remarques sur notre méthode 

1. En général, dans le cadre des variétés riemanniennes complètes à courbure de Ricci minorée, 

les estimations (1.2) n’offrent des informations précises que pour t → 0 + , et elles deviennent


très faibles lorsque t →+∞. Dans certains cas, la méthode de cet article nous permet d’obtenir 

des estimations optimales de type (1.2) lorsque t →+∞. 

Pour mieux comprendre les avantages de notre méthode (qui consiste à utiliser directement 

des estimations du noyau de la chaleur et de son gradient, plutôt que la formule de Bochner- 

Lichnerowicz, comme [4]), considérons une variété riemannienne compacte, sans bord et de 

dimension n, N. 

Notons pt le noyau de la chaleur sur N, λ1 > 0 la première valeur propre non nulle du Laplacien 

et |N| son volume. Alors pour t ≫ 1, on a d’une part : 

 

∇e t f(g) 

 

= 

∇ 

 

 

N 

N 

pt(g, g ′ 

) f(g ′ ) − 1 

|N| 

 

N 

N 

 

f(g∗)dμ(g∗) dμ(g ′ 

 

) 

 

 

∇pt(g, g ′ ) 

 

· 

f(g′ ) − 1 

 

 

 

f(g∗)dμ(g∗) 

|N| 

dμ(g′ ) 

C1e −λ1t 

 

 

 

f(g′ ) − 1 

 

 

 

f(g∗)dμ(g∗) 

|N| 

dμ(g′ ) 

N 

N 

par le fait que ∇pt(g, g ′ ) C1e−λ1t pour tout t ≫ 1etg,g ′ ∈ N 

C2e −λ1t 

 

diam(N) |∇f |(g ′ )dμ(g ′ ) 

N 

par l’inégalité de Poincaré 

Ce −λ1t 

 

diam(N)|N| pt(g, g ′ )|∇f |(g ′ )dμ(g ′ ) 

N 

par le fait que pt(g, g ′ ) 2 −1 |N| −1 pour tout t ≫ 1etg,g ′ ∈ N 

Ce −λ1t diam(N)|N| e t |∇f | w (g) 1/w , 

pour tout g ∈ N, f ∈ C ∞ (N) et 1 we 

∇υ1Lw −1/w 2 

, 

|N| 

puisque pt(g, g ′ ) 2|N| −1 pour tout t ≫ 1etg,g ′ ∈ N. 

Autrement dit, dans le cadre des variétés riemanniennes compactes, sans bord, on a 

sup 

g∈N,f∈C ∞ (N) 

|∇e t f(g)| 

[e t (|∇f | w )(g)] 1/w ∼ e−λ1t .


Et on remarque que lorsque t →+∞, le facteur e−λ1t est décroissant beaucoup plus vite que 

e− max(0,ko)t avec ko = sup{k ∈ R; Ric(N) k}, en rappelant que le théorème de Lichnerowicz 

nous dit que λ1 n 

n−1ko pour ko > 0. 

2. Dans [21,23], les estimations (1.4) avec 1


où 

 

f(g ′ ) − fr(g) C 

 

B(g,r) 

 

∇f(g∗) 

d(g ′ ,g∗) 

|B(g ′ ,d(g ′ ,g∗))| dg∗, ∀g ′ ∈ B(g,r), (2.2) 

fr(g) = B(g,r) −1 

 

B(g,r) 

f(g ′ )dg ′ . 

On remarque aussi que sur H 1 , l’inégalité de Poincaré (2.1) et la formule de représentation 

(2.2) sont équivalentes, voir [19] (ou bien voir [10,11] pour des résultats un peu faibles mais 

suffisants pour montrer notre résultat). 

3. Quelques résultats sur H 1 

3.1. Rappel sur la structure sous-riemannienne de H 1 

On rappelle dans cette partie l’expression explicite de la distance de Carnot–Carathéodory et 

de la géodésique minimale entre l’origine o et (x, t) ∈ H 1 . Presque tous les résultats de cette 

partie proviennent de [5, pp. 635–644] et [12]. 

Dans la suite, pour simplifier les notations, on suppose que 

Posons 

Alors 

x = (x1,x2) = (0, 0). 

2ϕ − sin 2ϕ 

μ(ϕ) = 

2sin2 : ]−π,π[→R. (3.1) 

ϕ 

ψ(x,ϕ)= x,μ(ϕ)x 2 : (x, ϕ) ∈ R 2 ×]−π,π[; x = (0, 0) → (x, t) ∈ H 1 ; x = (0, 0) 

est un C 1 -difféomorphisme. 

Notons μ −1 et ψ −1 la fonction réciproque de μ et de ψ respectivement, alors 

d 2 2 θ 

(x, t) = x 

sin θ 

2 

avec θ = μ −1 

 

t 

x2 

(voir (1.40) et (1.30) de [5], et on a posé a = τ = 1 et remplacé 2θ par θ dans (1.30)), et la 

géodésique minimale unique entre l’origine et (x, t), 

γ(s)= x1(s), x2(s), t(s) : [0, 1]→H 1 , 

s’écrit comme (il suffit de poser a = τ = 1 et de remplacer 2θ par θ dans le Theorem 1.21 de [5]) 

(3.2) 

(3.3)

x(s) = 


 

x1(s) 

= 

x2(s) 

sin sθ 

 

cos(s − 1)θ sin(s − 1)θ 

× 

sin θ − sin(s − 1)θ cos(s − 1)θ 

x1 

x2 

 

, (3.4) 

θ 

t(s)= t − (1 − s) 

sin2 θ x2 + 1 sin 2θ − sin 2sθ 

2 sin2 x 

θ 

2 

= 2sθ − sin 2sθ 

2sin2 x 

θ 

2 2sθ − sin 2sθ 

= 

2sin2 

x(s) 2 , 

sθ 

autrement dit, on a 

(3.5) 

μ −1 

 

t(s) 

x(s)2 

= sμ −1 

 

t 

x2 

, ∀0 s 1. (3.6) 

3.2. Estimations optimales du noyau de la chaleur et de son gradient 

Par [12,14] ou [20], on a l’expression explicite de ph comme suit : 

ph(x, t) = 

1 

2(4πh) 2 

 

R 

 

λ 2 

exp tı −x coth λ 

4h 

λ 

dλ. (3.7) 

sinh λ 

Pour montrer le Théorème 1.1, on aura besoin des estimations optimales de p(x,t) et de 

|∇p(x,t)| comme suit : 

Lemme 3.1. Il existe une constante L1 > 1 telle que pour tout (x, t) ∈ H 1 ,ona: 

L −1 

1 e−d2 (x,t)/4 1 +xd(x,t) −1/2 p(x,t) L1e −d2 (x,t)/4 1 +xd(x,t) −1/2 . (3.8) 

Lemme 3.2. Il existe une constante L2 > 1 telle que : 

 

∇p(x,t) L2d(x,t)p(x,t), ∀(x, t) ∈ H 1 . (3.9) 

Preuve du Lemme 3.1. L’estimation (3.8) est une conséquence immédiate de l’expression explicite 

de la distance de Carnot–Carathéodory (3.3) et des Theorems 1.1, 1.3 et Remark 1.5 de 

[13] (voir aussi [5,12] pour des résultats partiels). ✷ 

Preuve du Lemme 3.2. Observons que 

et 

X1p(x,t) = 

1 

4(4π) 2 

 

 

∇p(x,t) 2 =|X1p| 2 (x, t) +|X2p| 2 (x, t), 

R 

X2p(x,t) =− 1 

4(4π) 2 

 

λ 

2 

λ(−x1 coth λ + x2ı)exp tı −x coth λ 

4 

λ 

sinh λ dλ, 

 

R 

 

λ 

2 

λ(x2 coth λ + x1ı)exp tı −x coth λ 

4 

λ 

sinh λ dλ.


Soient 

Alors 

 

W1 = 

R 

R 

 

λ 

2 

λ exp tı −x coth λ 

4 

λ 

sinh λ dλ, 

 

λ 

2 

W2 = cosh λ exp tı −x coth λ 

4 

2 λ 

dλ. 

sinh λ 

 

∇p(x,t) 

1 

 

4(4π) 2 

 

|x1|+|x2| |W1|+|W2| . 

Comme |x1|+|x2| 2x 2d(x,t) (voir (3.3)), par (3.8), pour terminer la preuve de (3.9), 

il nous reste à montrer qu’il existe une constante C>0 telle que pour tout (x = (0, 0), t) ∈ H 1 , 

ona: 

|W1| Ce − d2 (x,t) −1/2, 4 1 +xd(x,t) (3.10) 

|W2| C d(x,t) 

x e− d2 (x,t) −1/2. 4 1 +xd(x,t) (3.11) 

En comparant l’expression intégrale de W1 

(resp. W2) avec celle de p (resp. p∗(x = 

x2 1 + x2 2 + x2 3 + x2 4 ,t) = p∗1(x1,x2,x3,x4,t), le noyau de la chaleur en temps h = 1surle 

groupe de Heisenberg de dimension réelle 5, H2 ), on trouve qu’il y a dans l’intégrale qu’une différence 

d’un facteur, la fonction analytique sur le plan complexe, (2π) −2λ (resp. (2π) −3 cosh λ). 

On observe aussi que l’estimation W1 est analogue à celle de p obtenue dans le Theorem 2.17 de 

[5] et que l’estimation W2 est plus faible que celle de p∗1 obtenue dans le Theorem 4.6 de [5]. 

Donc, il suffit de répéter mot par mot la preuve du Theorem 2.17 (resp. Theorem 4.6 avec n = 2) 

de [5] pour montrer (3.10) (resp. (3.11)). ✷ 

4. Preuve du Théorème 1.1 

Dans la suite, V(r)=|B(o,r)| (r >0). On rappelle que 

 

B(g,r) = V(r)∼ r 4 , ∀g ∈ H 1 ,r>0. (4.1) 

Par l’invariance à gauche de ∇ et de ph ainsi que la dilatation sur H 1 , on remarque que pour 

montrer le Théorème 1.1, il suffit de montrer qu’il existe une constante C>1 telle que (voir 

aussi Lemma 2.3 et Proposition 2.6 de [9] pour l’explication détaillée) : 

 

∇e f (o) Ce |∇f | (o), ∀f ∈ C ∞ o 

. (4.2)


Dans toute la suite, pour simplifier les notations, on pose 

1 

fo = 

|B(o,1)| 

 

B(o,1) 

f(g)dg, A∞ = e 1000 (L1 + L2) 8 , 

où L1,L2 > 1 proviennent de (3.8) et (3.9) respectivement. 

L’idée principale de la preuve de (4.2). On explique brièvement l’idée principale de la preuve 

de (4.2) : 

Comme H1 est stochastiquement complète, on a 

 

∇e f 

 

(o) = 

∇p g −1 

 

f(g)−fo dg 

 

 

 

∇p g −1 · 

f(g)−fo dg. 

H 1 

On écrit formellement 

 

∇e f 

 

(o) ∇p g −1 · 

f(g)−fo dg 

H 1 

 

 

H 1 

 

= 

H 1 

 

∇p g −1 

 

 

∇f(g ′ ) 

 

H 1 

H 1 

H 1 

 

C p(g ′ ) ∇f(g ′ ) dg ′ . 

H 1 

 

∇f(g ′ ) K(g,g ′ )dg ′ 

 

dg 

 

∇p g −1 K(g,g ′ )dg 

Autrement dit, pour démontrer l’estimation (4.2), il suffit de trouver une fonction K définie 

sur H 1 × H 1 qui satisfait les deux conditions suivantes : 

 

f(g)− fo 

 

H 1 

 

 

 

H 1 

 

∇f(g ′ ) K(g,g ′ )dg ′ , ∀g ∈ H 1 ,f∈ C ∞ o 

 

dg ′ 

H 1 , (4.3) 

 

∇p g −1 K(g,g ′ )dg Cp(g ′ ), ∀g ′ ∈ H 1 . ✷ (4.4) 

La condition (4.3) nous conduit naturellement à utiliser l’inégalité de Poincaré (localement) 

et la formule de représentation (localement). La difficulté se trouve à remplir la condition (4.4) : 

on doit choisir K à partir de (4.3) le mieux possible, et doit utiliser les estimations optimales 

du noyau de la chaleur et de son gradient ainsi que la structure sous-riemannienne de H 1 .On 

verra aussi que la difficulté pour contrôler K(g,g ′ ) se trouve dans le cas où d(g,g ′ ) ≪ 1avec 

d(g) →+∞.


Preuve de (4.2). Soient 

on a 

Par le fait que 

Comme 

J1 = 

J2 = 

 

{g∈H 1 ;d(g)A∞} 

 

{g∈H 1 ;d(g)>A∞} 

d(g)p(g) · 

f(g)−fo dg, 

d(g)p(g) · 

f(g)−fo dg. 

 

∇p g −1 L2d g −1 p g −1 = L2d(g)p(g), 

e |∇f | 

(o) = 

 

∇e f (o) L2(J1 + J2). 

H 1 

p g −1 

|∇f |(g) dg = 

H 1 

p(g)|∇f |(g) dg, 

pour démontrer (4.2), il nous reste à montrer qu’il existe une constante C>0, qui ne dépend pas 

de f , telle que 

 

J1 + J2 C p(g)|∇f |(g) dg. ✷ 

4.1. Estimation de J1 

On voit que 

 

f(g)− fo 

H 1 

 

 

f1(g) − fo 

+ f(g)−f1(g) . 

La formule de représentation (voir (2.2)) nous dit que 

Par ailleurs, 

J11∗ = f(g)− f1(g) C1 

= C1 

 

B(g,1) 

 

B(g,1) 

 

∇f(g ′ ) 

d(g,g ′ ) 

|B(g,d(g,g ′ ))| dg′ 

 

∇f(g ′ ) 

d(g ′ ,g) 

|B(g ′ ,d(g ′ ,g))| dg′ , par (4.1). 

J12∗ = fo − f1(g) fo − fA∞+2(o) + f1(g) − fA∞+2(o) .


En utilisant l’inégalité de Poincaré (voir (2.1)), on a 

et de la même façon, 

 

fo − fA∞+2(o) = 

 

1 

 

|B(o,1)| 

1 

 

|B(o,1)| 

C(A∞) 

 

B(o,1) 

 

B(o,A∞+2) 

 

B(o,A∞+2) 

 

f1(g) − fA∞+2(o) C(A∞) 

′ 

f(g ) − fA∞+2(o) dg ′ 

 

 

 

 

 

f(g ′ ) − fA∞+2(o) dg ′ 

|∇f |(g ′ )dg ′ , 

 

B(o,A∞+2) 

Par les estimations supérieures de p, (3.8), on a donc 

J1 C ′ (A∞) 

C∗(A∞) 

C(A∞) 

C ∗ (A∞) 

 

B(o,A∞+1) 

 

B(o,A∞+2) 

 

B(o,A∞+2) 

 

B(o,A∞+2) 

 

B(g,1) 

 

∇f(g ′ ) 

d(g ′ ,g) 

|B(g ′ ,d(g ′ ,g))| dg′ + 

|∇f |(g ′ 

) 1 + 

|∇f |(g ′ )dg ′ 

 

B(g ′ ,1) 

p(g ′ )|∇f |(g ′ )dg ′ 

C ∗ 

(A∞) p(g ′ )|∇f |(g ′ )dg ′ . 

H 1 

4.2. Estimation de J2 

Puisque 

J2 = 

on peut supposer dans toute la suite 

 

par (4.1) 

{g=(x,t)∈H 1 ;d(g)>A∞,x=(0,0)} 

|∇f |(g ′ )dg ′ . 

d(g ′ ,g) 

|B(g ′ ,d(g ′ ,g))| dg 

 

par (3.8) 

 

B(o,A∞+2) 

dg ′ 

d(g)p(g) 

f(g)−fo dg, 

g = (x, t) ∈ Σ = R 2 × R \ (0, 0,t); t ∈ R 

; d(g) > A∞ . 

|∇f |(g ′ )dg ′ 

 

dg


Notons γ la géodésique minimale (unique) d’origine à g = (x, t), qui est définie par (3.4) 

et (3.5). 

Soit 

N(g)= 100d 3/2 (g) ln d(g) + 1, 

où [h] (h ∈ R) désigne la partie entière de h,etsoit 

g N(g) = γ 1 − N(g)d −7/2 (g) . 

On doit dire au lecteur que le choix de N(g) et {g(i) = γ(1− id−7/2 (g))}0id(g)N(g) défini 

dans la Section 4.2.2 n’est pas unique mais très délicat, ça dépend non seulement des estimations 

optimales du noyau de la chaleur et de son gradient et aussi de la structure sous-riemannienne 

de H1 (plus précisment, on aura besoin des résultats analogues aux Lemmes 4.2 et 4.3 de cet 

article). 

Alors 

 

f(g)−fo 

f2d−5/2 (g) g N(g) − fo 

+ 

f(g)−f2d−5/2 (g) g N(g) . 

En utilisant l’inégalité de Poincaré (voir (2.1)) et l’estimation du volume des boules dans H1 (voir (4.1)), on a 

 

f2d−5/2 (g) g N(g) − fo 

fo − fd(g(N(g)))+4d−5/2 (g) (o) 

+ 

f2d−5/2 (g) g N(g) − fd(g(N(g)))+4d−5/2 (g) (o) 

Cd 11 

 

(g) 

∇f(g ′ ) dg ′ . 

Donc, 

 

f(g)−fo 11 

Cd (g) 

Soient 

J21 = 

 

J22 = 

On constate que 

 

 

B o,d(g)−90 

{g∈H 1 ;d(g)>A∞} 

Σ 

ln d(g) 

d(g) 

B o,d(g)−90 

ln d(g) 

d(g) 

 

′ 

∇f(g ) ′ 

dg + f(g)−f2d−5/2 (g) g N(g) . 

d 12 

(g)p(g) 

ln d(g) 

B(o,d(g)−90 d(g) ) 

d(g)p(g) 

f(g)−f2d−5/2 (g) g N(g) dg. 

J2 C(J21 + J22). 

 

∇f(g ′ ) dg ′ 

 

dg,


Pour achever la preuve du Théorème 1.1, il suffit de montrer qu’il existe une constante C>0 

telle que pour toute f ∈ C ∞ o (H1 ),ona 

 

J21 + J22 C 

4.2.1. Estimation de J21 

Par un calcul simple, on observe que 

H 1 

p(g) ∇f(g) dg. 

 

(g, g ′ ) ∈ H 1 × H 1 ; d(g) > A∞,d(g ′ 

ln d(g) 

) A∞,d(g ′ ) A∞ − 1 

 

∪ (g, g ′ ) ∈ H 1 × H 1 ; d(g ′ )>A∞ − 1,d(g)>d(g ′ ) + 46 ln d(g′ ) 

d(g ′ 

. 

) 

Donc, en utilisant les estimations supérieures de p(g) (voir (3.8)), on a 

J21 C 

 

d(g ′ )A∞−1 

+ C 

 

d(g ′ )>A∞−1 

 

∇f(g ′ ) dg ′ 

 

d(g)>A∞ 

 

∇f(g ′ ) 

 

d 12 (g)e −d2 (g)/4 dg 

d(g)>d(g ′ )+46 ln d(g′ ) 

d(g ′ ) 

Par l’estimation du volume des boules dans H 1 (voir (4.1)), on a 

 

d(g)>A∞ 

et lorsque d(g ′ )>A∞ − 1, on a 

d 12 (g)e −d2 

(g)/4 

dg dg ′ . 

d 12 (g)e −d2 (g)/4 14 

dg CA∞e −A2∞ /4 C 1 + A 2 −1/2e−(A∞−1) ∞ 

2 /4 

, 

 

d(g)>d(g ′ )+46 ln d(g′ ) 

d(g ′ ) 

d 12 (g)e − d2 (g) 

4 dg Cd −1 (g ′ )e − d2 (g ′ ) 

4 . 

Par conséquent, les estimations inférieures de p (voir (3.8)) nous disent que 

 

J21 C 

H 1 

 

∇f(g ′ ) p(g ′ )dg ′ .


4.2.2. Estimation de J22 

Rappelons que 

A∞ = e 1000 (L1 + L2) 8 , 

où les deux constantes L1,L2 > 1 proviennent de (3.8) et (3.9) respectivement. 

Rappelons aussi que 

où 

 

J22 = 

Σ 

d(g)p(g) 

f(g)−f2d−5/2 (g) g N(g) dg, 

Σ = R 2 × R \ (0, 0,t); t ∈ R 

; d(g) > A∞ , 

N(g)= 100d 3/2 (g) ln d(g) + 1, g N(g) = γ 1 − N(g)d −7/2 (g) , 

et γ désigne la géodésique minimale (unique) d’origine à g = (x, t), qui est définie par (3.4)– 

(3.6). 

La difficulté de la preuve du Théorème 1.1 se trouve à montrer qu’il existe une constante 

C>0 telle que pour toute f ∈ C∞ o (H1 ),ona 

 

J22 C p(g) ∇f(g) dg. (4.5) 

Dans la suite, pour simplifier les notations, on note 

Et on remarque que 

On estime maintenant 

Observons que 

N(g)−1 

J22∗ 

H 1 

g = (x, t) ∈ Σ, 

s(i) = 1 − id −7/2 (g), pour 0 i d(g)N(g), 

g(i) = x(i),t(i) = x1(i), x2(i), t(i) = γ s(i) . 

d g(i),g(j) =|i − j|d −5/2 (g), ∀0 i, j d(g)N(g). 

i=0 

J22∗ = 

f(g)−f2d−5/2 (g) g N(g) . 

 

f g(i) − f g(i + 1) + f g N(g) 

− f2d−5/2 (g) g N(g) . 

Par la formule de représentation (voir (2.2)), on a

et 

et 

Donc, 


 

f g N(g) − f2d−5/2 (g) g N(g) 

 

 

C 

∇f(g ′ ) 

d(g(N(g)),g ′ ) 

|B(g(N(g)),d(g(N(g)),g ′ ))| dg′ , 

B(g(N(g)),4d −5/2 (g)) 

 

f g(i) − f g(i + 1) 

f g(i) 

− f2d−5/2 (g) g(i) + f g(i + 1) 

− f2d−5/2 (g) g(i) 

 

 

C 

∇f(g ′ ) 

d(g(i),g ′ ) 

|B(g(i), d(g(i), g ′ ))| dg′ 

 

J22 C 

B(g(i),4d −5/2 (g)) 

+ C 

 

B(g(i+1),4d −5/2 (g)) 

N(g) 

J22∗ C 

Σ 

 

i=0 

B(g(i),4d−5/2 (g)) 

N(g) 

 

d(g)p(g) 

i=0 

 

′ 

∇f(g ) 

d(g(i + 1), g ′ ) 

|B(g(i + 1), d(g(i + 1), g ′ ))| dg′ . 

B(g(i),4d −5/2 (g)) 

 

∇f(g ′ ) 

d(g(i),g ′ ) 

|B(g(i), d(g(i), g ′ ))| dg′ , (4.6) 

 

∇f(g ′ ) 

d(g(i),g ′ ) 

|B(g(i), d(g(i), g ′ ))| dg′ 

 

dg. 

Avant de continuer la preuve de (4.5), on doit dire au lecteur que l’étape crucial pour démontrer 

(4.5) (et donc le Théorème 1.1) se trouve à obtenir l’estimation (4.6) et que le choix 

de {g(i)}0ig(N) et d −5/2 (g) dans l’estimation (4.6) n’est pas unique mais très délicat (il dépend 

des estimations optimales du noyau de la chaleur et de son gradient ainsi que la structure 

sous-riemannienne de H 1 ). 

Notons χ la fonction caractéristique, 

Q = Q(g, g ′ N(g) 

) = d(g) 

i=0 

d(g ′ , g(i)) 

|B(g(i),d(g ′ , g(i)))| χ 

 

(g, g ′ ) ∈ Σ × H 1 ; d g ′ ,g(i) < 

J ∗ 22 = 

 

Alors, en utilisant le théorème de Fubini, on a 

 

J22 C 

Σ 

p(g)Qdg. 

d(g ′ )>A∞−100(ln A∞)/A∞−5A −5/2 

∞ 

 

∇f(g ′ ) J ∗ 22 dg′ . 

4 

d5/2 

, 

(g)


Donc, pour montrer (4.5), il nous reste à montrer le résultat suivant : 

Proposition 4.1. Il existe une constante C>0 telle qu’on a 

Preuve. Observons que 

J ∗ 22 Cp(g′ ), ∀d(g ′ ln A∞ 

)>A∞ − 100 − 5A 

A∞ 

−5/2 

∞ . (4.7) 

Q = 0 ⇒ d g ′ ,g(i) 4 

< 

d5/2 (g) 

pour un certain 0 i N(g)+ 1 

⇒ 

d(g ′ ) + 4/d5/2 (g) 

1 − 200(ln d(g))/d2 d(g(i)) 

d(g) = 

(g) s(i) d(g′ 4 

) − 

d5/2 (g) 

⇒ d(g′ )/d(g) + 4/d7/2 (g) 

1 − 200(ln d(g))/d2 (g) 1 d(g′ ) 

d(g) − 

4 

d7/2 . 

(g) 

(4.8) 

En particulier, on a 

N(g ′ ) − 1 N(g) N(g ′ ) + 1, et 

En utilisant (4.1), on a donc 

Q 2d(g ′ N(g 

) 

′ )+1 

Par conséquent, 

Rappelons que 

i=0 

J ∗ 22 2d(g′ N(g 

) 

′ )+1 

2 

√ d(g) d(g 

5 ′ √ 

5 

) d(g). (4.9) 

2 

d(g ′ , g(i)) 

|B(g ′ ,d(g ′ , g(i)))| χ 

 

(g, g ′ ) ∈ Σ × H 1 ; d g ′ ,g(i) < 

i=0 

 

{g∈Σ;d(g ′ ,g(i))


Lemme 4.2. Soit L1 > 1 comme dans (3.8) et soit (g = (x, t), g ′ ) ∈ Σ × H 1 satisfaisant 

alors 

d g ′ ,g(i) < 4d −5/2 (g) pour un certain 0 i N(g ′ ) + 1, 

p(g) 400L 2 1p(g′ )e −i/(5d3/2 (g ′ 

)) sin s(i)μ−1 (t/x2 ) 

sin μ−1 (t/x2 1/2 . (4.11) 

) 

Lemme 4.3. Il existe une constante C>0 telle que pour tout g ′ ∈ H 1 et tout 0 i N(g ′ ) + 1, 

on a 

 

{g=(x,t)∈Σ;d(g ′ ,g(i))< 4 

d 5/2 (g) } 

sin s(i)μ −1 (t/x 2 ) 

sin μ −1 (t/x 2 ) 

1/2 

d(g ′ , g(i)) 

|B(g ′ ,d(g ′ , g(i)))| dg 

Cd −5/2 (g ′ ). (4.12) 

On admet les deux lemmes pour l’instant et on continue avec la démonstration de la Proposition 

4.1. 

Fin de la preuve de la Proposition 4.1. Par (4.10)–(4.12), on a 

J ∗ 22 Cd−3/2 (g ′ )p(g ′ N(g 

) 

′ )+1 

e − i 5 d−3/2 (g ′ 

) ′ 

Cp(g ) 1 + 

D’où la Proposition 4.1. ✷ 

i=0 

+∞ 

0 

e −h/5 

dh = 6Cp(g ′ ). 

Preuve du Lemme 4.2. Par les estimations supérieures et inférieures de p (voir (3.8)), on a 

et 

p(g) = p g(i) p(g) 

p(g(i)) L21 pg(i) e d2 (g(i))−d2 (g) 

4 

Et par la définition de g(i) et (3.5), on a 

d g(i) d(g), 

1 +x(s(i))d(g(s(i))) 

1 +xd(g) 

 

x s(i) = sin s(i)μ−1 (t/x 2 ) 

sin μ −1 (t/x 2 ) x, 

e d2 (g(i))−d 2 (g) 

4 = e − d2 (g) 

4 (1−s(i) 2 )


donc, 

p(g) 100L 2 1pg(i) e − i 5 d−3/2 (g ′ 

) sin s(i)μ−1 (t/x2 ) 

sin μ−1 (t/x2 1/2 . 

) 

Et pour terminer la preuve du Lemme 4.2, il nous reste à montrer que 

En fait, on va montrer que 

p g(i) 4p(g ′ ). 

 

 

 

p(g(i)) 

p(g ′ ) 

 

 

− 1 

1. (4.13) 

Preuve de (4.13). On observe que 

 

 

 

p(g(i)) 

p(g ′ 

 

− 1 

) = d(g′ , g(i)) 

p(g ′ 

∇p(g∗) 

) 

avec g∗ ∈ B g ′ , 4d−5/2 (g) . 

Mais, par les estimations de p et de |∇p| (voir (3.8) et (3.9)), on a 

1 

p(g ′ 1 d 

2 ′ 

L1 1 + d (g ) 2 e 

) 2 (g ′ ) 

4 2L1d(g ′ )e d2 (g ′ ) 

4 , 

 

∇p(g∗) 

′ −5/2 − 

L2d(g∗)p(g∗) L1L2 d(g ) + 4d (g) e (d(g′ )−4d−5/2 (g)) 2 

4 

4L1L2d(g ′ )e − d2 (g ′ ) 

4 , 

par (4.9) et le fait que d(g) > 1000. 

D’ailleurs, (4.9) et le fait que g ∈ Σ nous disent que 

On a donc (4.13). ✷ 

d g ′ ,g(i) < 4d −5/2 (g) < 5d −2 (g)A −1/2 

∞ 

< 8L 2 1L2 −1d −2 ′ 

(g ). 

Preuve du Lemme 4.3. Par (4.9), on peut supposer que d(g ′ )> 1 2 A∞. 

Rappelons que 

et que 

2ϕ − sin 2ϕ 

μ(ϕ) = 

2sin2 : ]−π,π[→R, 

ϕ 

ψ(x,ϕ)= x,μ(ϕ)x 2 : (x, ϕ) ∈ R 2 ×]−π,π[; x = (0, 0) → (x, t) ∈ H 1 ; x = 0 

est un C 1 -difféomorphisme. Notons Dψ(x,θ) la matrice jacobienne de ψ en (x, θ), ona

et 

Posons 


2sinθ − 2θ cos θ 

det Dψ(x,θ) = 

sin3 x 

θ 

2 . 

 

Ωg ′ = g = (x, t) ∈ H 1 ; x = (0, 0), d(g) > 1 

2 d(g′ 

) . 

Pour 0 i N(g ′ ) + 1, définissons 

Θi(g) = g(i): Ωg ′ → H1 \ (0, 0,t)∈ H 1 ; t ∈ R , 

Φi : ψ −1 (Ωg ′) → R 2 \ (0, 0) ×]−π,π[, 

(x, θ) = ψ −1 g = (x, t) ↦→ ψ −1 g(i) = x(i), s(i)θ , 

où (voir la définition de s(i), (3.3) et (3.4)) 

i 

s(i) = s(i)(g) = 1 − 

d7/2 (g) 

avec d(g) = θ 

 

t 

x, θ= μ−1 

sin θ x2 

, (4.14) 

sin s(i)θ 

x1(i) = x1 cos 

sin θ 

s(i) − 1 θ + x2 sin s(i) − 1 θ , (4.15) 

sin s(i)θ 

x2(i) = −x1 sin 

sin θ 

s(i) − 1 θ + x2 cos s(i) − 1 θ . (4.16) 

Observons que Φi et Θi sont de classe C 1 , et que 

Θi = ψ ◦ Φi ◦ ψ −1 . 

Et on a besoin des deux résultats suivants dont la preuve sera donnée plus tard : 

(i) Φi est injective sur ψ−1 (Ωg ′) ; 

(ii) on a 

 

det DΦi (x, θ) = 

sin s(i)θ 

sin θ 

2 1 + 5 

2 id−7/2 

(g) . (4.17) 

Les deux résultats précédents nous disent que Θi est aussi injective sur Ωg ′, et pour tout 

g = ψ(x,θ)∈ Ωg ′,ona 

det DΘi(g) = det Dψ x(i), s(i)θ · det DΦi(x, θ) · det Dψ(x,θ) −1 

sin s(i)θ 

4 

sin θ − θ cos θ 

= 

sin θ sin3 −1 sin s(i)θ − s(i)θ cos s(i)θ 

θ 

sin3 

1 + 

s(i)θ 

5 i 

2 d7/2 

= 0, 

(g)


donc, en distinguant les deux cas, |θ|=μ −1 (|t|/x 2 ) π/2 etπ/2 < |θ|

⎛ 

⎜ 

⎝ − 


sin s(i)θ 

sin θ K1 + ∂s(i) dx1(i) 

∂x1 ds(i) 

sin s(i)θ 


∂x1 ds(i) 

θ ∂s(i) 

∂x1 

sin s(i)θ 


∂x2 ds(i) 

sin s(i)θ 


∂x2 ds(i) 

θ ∂s(i) 

∂x2 

∂s(i) dx1(i) dx1(i) 

∂θ ds(i) + dθ 

∂s(i) dx2(i) dx2(i) 

∂θ ds(i) + dθ 

s(i) + θ ∂s(i) 

∂θ 

où le sens de dx1(i)/ds(i), dx1(i)/dθ, dx2(i)/ds(i), dx2(i)/dθ est claire. 

Soient 

R ∗ 1 = 

 

 

 

 

− 

R ∗ 2 = 

 

 

 

 

 

− 

 

 

sin s(i)θ 


∂x1 ds(i) 

sin s(i)θ 


∂x1 ds(i) 

sin s(i)θ 

sin θ K1 

sin s(i)θ 

sin θ K2 

∂s(i) 

∂x1 

sin s(i)θ 


∂x2 ds(i) 

sin s(i)θ 


∂x2 ds(i) 

sin s(i)θ 

sin θ K2 

sin s(i)θ 

sin θ K1 

∂s(i) 

∂x2 

R1 = s(i)R ∗ 1 , R2 = θR ∗ 2 . 

dx1(i) 

dθ 

dx2(i) 

dθ 

∂s(i) 

∂θ 

Par le fait que 

 

θ ∂s(i) 

,θ 

∂x1 

∂s(i) 

,s(i)+ θ 

∂x2 

∂s(i) 

 

= 

∂θ 

0, 0,s(i) 

∂s(i) 

+ θ , 

∂x1 

∂s(i) 

, 

∂x2 

∂s(i) 

 

, 

∂θ 

on peut donc écrire 

où on a noté 

 

 

 

 

R∗∗ = 

− 

 

 

sin s(i)θ 


∂x1 ds(i) 

sin s(i)θ 


∂x1 ds(i) 

∂s(i) 

∂x1 

det DΦi(x, θ) = s(i)R ∗ 1 + θR∗∗, 

sin s(i)θ 


∂x2 ds(i) 

sin s(i)θ 


∂x2 ds(i) 

∂s(i) 

∂x2 

 

 

 

 

 

, 

 

 

 

 

 

 

, 

∂s(i) dx1(i) 

∂θ ds(i) 

∂s(i) dx2(i) 

∂θ ds(i) 

∂s(i) 

∂θ 

⎞ 

⎟ 

⎠ 

 

dx1(i) 

+ 

dθ 

dx2(i) 

+ 

dθ . 

 

 

Or, la première ligne de R∗∗ peut s’écrire comme 

 

sin s(i)θ sin s(i)θ dx1(i) 

K1, K2, + 

sin θ sin θ dθ 

dx1(i) 

 

∂s(i) 

, 

ds(i) ∂x1 

∂s(i) 

, 

∂x2 

∂s(i) 

 

, 

∂θ 

et la seconde ligne de R∗∗ peut s’écrire comme 

 

− 

sin s(i)θ 

sin θ 

on a R∗∗ = R∗ 2 , donc 

 

sin s(i)θ dx2(i) 

K2, K1, + 

sin θ dθ 

dx2(i) 

 

∂s(i) 

ds(i) ∂x1 

, ∂s(i) 

, 

∂x2 

∂s(i) 

 

, 

∂θ 

det DΦi(x, θ) = R1 + R2. (4.18)


Calcul de R1. Puisque la première colonne de R∗ 1 peut s’écrire comme 

et la seconde colonne de R ∗ 1 

on a 

où on a noté 

on a 

sin s(i)θ 

sin θ 

 

K1 

+ 

−K2 

∂s(i) 

∂x1 

peut s’écrire comme 

sin s(i)θ 

sin θ 

K2 

K1 

 

+ ∂s(i) 

∂x2 

 

dx1(i)/ds(i) 

, 

dx2(i)/ds(i) 

 

dx1(i)/ds(i) 

, 

dx2(i)/ds(i) 

R ∗ 1 = 

2 

sin s(i)θ K1 K2 

 

sin θ −K2 K1 

+ sin s(i)θ 

 

∂s(i) 

 

sin θ ∂x2 

K1 

−K2 

 

dx1(i)/ds(i) 

 

∂s(i) 

dx2(i)/ds(i) + 

∂x1 

dx1(i)/ds(i) 

dx2(i)/ds(i) 

 

K2 

 

K1 

2 sin s(i)θ 

= 

+ 

sin θ 

sin s(i)θ 

K1R 

sin θ 

∗ 11 + K2R ∗ 

12 , 

R ∗ 11 

∂s(i) dx2(i) 

= 

∂x2 ds(i) 

∂s(i) dx1(i) 

+ 

∂x1 ds(i) , R∗ 12 

∂s(i) dx1(i) 

= 

∂x2 ds(i) 

− ∂s(i) 

∂x1 

Il nous reste à calculer ∂s(i)/∂x1, ∂s(i)/∂x2, dx1(i)/ds(i) et dx2(i)/ds(i). 

Dans la suite, pour simplifier les notations, on pose 

s ′ (i) = 

d 7 

s(i) = 

d(d(g)) 2 id−9/2 (g). 

On commence par calculer ∂s(i)/∂x1 et ∂s(i)/∂x2. Par le fait que (voir (3.3)) 

d 2 

θ 

2x 

2 

(g) = 

1 + x 

sin θ 

2 2 , 

dx2(i) 

. (4.19) 

ds(i) 

∂s(i) 

= s 

∂x1 

′ 1 ∂d 

(i) 

2d(g) 

2 (g) 

= s 

∂x1 

′ (i) 1 

2 θ 

x1, 

d(g) sin θ 

(4.20) 

∂s(i) 

= s 

∂x2 

′ 1 ∂d 

(i) 

2d(g) 

2 (g) 

= s 

∂x2 

′ (i) 1 

2 θ 

x2. 

d(g) sin θ 

(4.21) 

On calcule maintenant dx1(i)/ds(i) et dx2(i)/ds(i). Par (4.15), (4.16) et des formules élémentaires 

liant les fonctions trigonométriques, on peut écrire

Donc, 


x1(i) = 1 

x1 sin θ + sin 2s(i) − 1 θ + x2 cos θ − cos 2s(i) − 1 θ , 

2sinθ 

(4.22) 

x2(i) = 1 

−x1 cos θ − cos 2s(i) − 1 θ + x2 sin θ + sin 2s(i) − 1 θ . 

2sinθ 

(4.23) 

dx1(i) 

ds(i) 

dx2(i) 

ds(i) 

θ 

= x1 cos 

sin θ 

2s(i) − 1 θ + x2 sin 2s(i) − 1 θ , (4.24) 

θ 

= −x1 sin 

sin θ 

2s(i) − 1 θ + x2 cos 2s(i) − 1 θ . (4.25) 

Par (4.19)–(4.21), (4.24) et (4.25), on constate d’abord que 

R ∗ 11 = s′ (i) 1 

 

θ 

d(g) 

sin θ 

R ∗ 12 = s′ (i) 1 

 

θ 

d(g) sin θ 

3 

x 2 cos 2s(i) − 1 θ, 

3 

x 2 sin 2s(i) − 1 θ, 

puis, en utilisant le fait que d 2 (g) = ( θ 

sin θ )2 x 2 (voir (3.3)), que 

Donc, 

R ∗ 11 = s′ (i)d(g) θ 

sin θ cos 2s(i) − 1 θ, 

R ∗ 12 = s′ (i)d(g) θ 

sin θ sin 2s(i) − 1 θ. 

K1R ∗ 11 + K2R ∗ 12 

= s ′ (i)d(g) θ 

cos s(i) − 1 θ cos 2s(i) − 1 θ + sin s(i) − 1 θ sin 2s(i) − 1 θ 

sin θ 

= s ′ (i)d(g) θ 

cos s(i)θ. 

sin θ 

Par conséquent, 

Calcul de R2. Soient 

 

sin s(i)θ 

R1 = s(i) 

sin θ 

2 sin s(i)θ 

+ s(i) s 

sin θ 

′ (i)d(g) θ 

cos s(i)θ. (4.26) 

sin θ 

R ∗ 21 = 

 

sin s(i)θ 

sin θ 

2 ∂s(i) 

∂θ ,


R ∗ 22 

 

 

sin s(i)θ dx1(i) ∂s(i) ∂s(i) 

= −K2 − K1 − 

sin θ dθ ∂x2 ∂x1 

dx2(i) 

 

 

∂s(i) ∂s(i) 

K1 − K2 

dθ ∂x2 ∂x1 

= sin s(i)θ 

 

dx2(i) ∂s(i) 

K2 

− 

sin θ dθ ∂x1 

dx1(i) 

 

∂s(i) dx1(i) ∂s(i) 

− K1 

+ 

dθ ∂x2 dθ ∂x1 

dx2(i) 

 

∂s(i) 

. 

dθ ∂x2 

On développe suivant la troisième colonne de R ∗ 2 , et on constate que R∗ 2 = R∗ 21 + R∗ 22 

R2 = θR ∗ 21 + θR∗ 22 . 

En rappelant que d(g) = θ 

sin θ x,ona 

donc, 

∂s(i) 

∂θ = s′ (i) ∂d(g) 

∂θ = s′ (i) 

R ∗ 21 = 

 

sin s(i)θ 

sin θ 


sin 2 θ 

2 

s ′ (i)d(g) 1 

θ 

x=s ′ (i)d(g) 1 

θ 


. 

sin θ 


, 

sin θ 

Pour calculer R∗ dx1(i) 

22 , on aura besoin de l’expression explicite de dθ et de dx2(i) 

dθ . Notons 

U1 = d sin θ + sin(2s(i) − 1)θ 

= 

dθ 2sinθ 

2s(i)sin θ cos(2s(i) − 1)θ − sin 2s(i)θ 

U2 = d cos θ − cos(2s(i) − 1)θ 

= 

dθ 2sinθ 

−1 + 2s(i)sin θ sin(2s(i) − 1)θ + cos 2s(i)θ 

Alors, par (4.22) et (4.23), 

dx1(i) 

dθ = x1U1 + x2U2, 

En utilisant (4.20) et (4.21), on constate que 

2sin 2 θ 

2sin 2 θ 

dx2(i) 

dθ =−x1U2 + x2U1. 

, 

. 

et donc 

R ∗ sin s(i)θ 

22 = s 

sin θ 

′ (i) 1 

2 θ 

x 

d(g) sin θ 

2 (−K2U2 − K1U1) 

sin s(i)θ 

=− s 

sin θ 

′ (i)d(g)(K1U1 + K2U2) par (3.3) 

sin s(i)θ 

=− s 

sin θ 

′ s(i)sin θ cos s(i)θ − cos θ sin s(i)θ 

(i)d(g) 

sin2 , 

θ 

où on a utilisé trois fois les formules élémentaires liant les fonctions trigonométriques dans la 

dernière égalité. 

On a donc 

R2 = θR ∗ 21 + θR∗ 22 = s′ 

sin s(i)θ 

(i)d(g) 

sin θ 

Par (4.18), (4.26) et (4.27), on obtient (4.17). ✷ 

2 sin s(i)θ 

− θ 

sin θ 

 

s(i)cos s(i)θ 

. (4.27) 

sin θ

5. Quelques problèmes ouverts 


Pour terminer cet article, on donne quelques questions intéressantes. 

1. J’espère que le lecteur peut suivre la méthode de la preuve du Théorème 1.1 pour obtenir un 

résultat analogue au Théorème 1.1 dans le cadre des groupes de Heisenberg de dimension 

réelle 5 et aussi dans le cadre des cônes classiques, C(Sϖ ) avec 0 0, f∈ C ∞ o (M). 

En particulier, on s’intéressait à obtenir des estimations du noyau de la chaleur et de son 

gradient à partir d’une telle inégalité. 

3. Sur les variétés coniques, C(N), à cause de la singularité conique, on trouve qu’ils ont beaucoup 

de propriétés spéciales par rapport aux variétés riemanniennes complètes à courbure 

de Ricci minorée, voir par exemple [8,15–17] ou [18]. Lorsque la première valeur propre 

non nulle du Laplacien sur N, λ1 dim N, ce sera très intéressant de savoir pour quels 

1 w 0, f∈ Co . 

Remarquons que dans cette situation, la courbure de Ricci sur C(N) peut être non minorée. 

Et on rappelle aussi que pour λ1 < dim N, les estimations précédentes ne sont pas valables 

(voir [17] ou [18]). 

4. Dans l’étude de la formule de représentation, ce sera très intéressant de savoir si la condition 

technique (H2) dans [19], i.e. en supposant que le volume des boules est au moins à 

croissance linéaire, est nécessaire. En tout cas, cette condition est faible. 

Remerciements 

L’auteur est soutenu par le DFG (SFB 611). Je tiens à remercier T. Coulhon et L. Saloff-Coste 

pour des discutions et de nombreuses suggestions, F. Lust-Piquard et M. Villani pour m’avoir 

communiqué [21]. Je voudrais remercier aussi le référé pour m’avoir indiqué une faute dans la 

version originaire de cet article et des suggestions. 

Références 

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Sobolev logarithmiques, in : Panoramas et Synthèses, vol. 10, Soc. Math. France, Paris, 2000. 

[2] P. Auscher, T. Coulhon, X.T. Duong, S. Hofmann, Riesz transform on manifolds and heat kernel regularity, Ann. 

Sci. École Norm. Sup. (4) 37 (2004) 911–957. 

[3] D. Bakry, Transformations de Riesz pour les semi-groupes symétriques. II. Étude sous la condition Γ2 0, in : 

Séminaire de probabilités, XIX, 1983/84, in : Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 145– 

174. 

[4] D. Bakry, Un critère de non-explosion pour certaines diffusions sur une variété riemannienne complète, C. R. Acad. 

Sci. Paris Sér. I Math. 303 (1986) 23–26.


[5] R. Beals, B. Gaveau, P.C. Greiner, Hamilton–Jacobi theory and the heat kernel on Heisenberg groups, J. Math. Pures 

Appl. (9) 79 (2000) 633–689. 

[6] J. Cheeger, M. Gromov, M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, 

and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982) 15–53. 

[7] T. Coulhon, X.-T. Duong, Riesz transform and related inequalities on noncompact Riemannian manifolds, Comm. 

Pure Appl. Math. 56 (2003) 1728–1751. 

[8] T. Coulhon, H.-Q. Li, Estimations inférieures du noyau de la chaleur sur les variétés coniques et transformées de 

Riesz, Arch. Math. 83 (2004) 229–242. 

[9] B.K. Driver, T. Melcher, Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal. 221 (2005) 

340–365. 

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vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995) 577–604. 

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spaces of homogeneous type, Int. Math. Res. Not. (1996) 1–14. 

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nilpotents, Acta Math. 139 (1977) 95–153. 

[13] H. Hueber, D. Müller, Asymptotics for some Green kernels on the Heisenberg group and the Martin boundary, 

Math. Ann. 283 (1989) 97–119. 

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of certain subelliptic operators on the Heisenberg group, Studia Math. 56 (1976) 165–173. 

[15] H.-Q. Li, La transformation de Riesz sur les variétés coniques, J. Funct. Anal. 168 (1999) 145–238. 

[16] H.-Q. Li, Estimations du noyau de la chaleur sur les variétés coniques et ses applications, Bull. Sci. Math. 124 

(2000) 365–384. 

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Paris 337 (2003) 283–286. 

[18] H.-Q. Li, Sur la continuité de Hölder du semi-groupe de la chaleur sur les variétés coniques, soumis. 

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Math. Soc. 30 (1998) 578–584. 

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233–249. 

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Melcher, prépublication. 

[22] P. Maheux, L. Saloff-Coste, Analyse sur les boules d’un opérateur sous-elliptique, Math. Ann. 303 (1995) 713–740. 

[23] T. Melcher, Hypoelliptic heat kernel inequalities on Lie groups, PhD thesis, 2004. 

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Pure Appl. Math. 58 (2005) 923–940. 

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vol. 100, Cambridge Univ. Press, Cambridge, 1992.


Large Fredholm triples ✩ 

Dan Kucerovsky 

Department of Mathematics and Statistics, UNB-F, Fredericton, NB, Canada E3B 5A3 

Received 28 July 2005; accepted 15 November 2005 

Available online 24 April 2006 

Communicated by Alain Connes 


Abstract 

We give several equivalent condition for Busby extensions of a given algebra to be absorbing, considerably 

improving our earlier results [G.A. Elliott, D. Kucerovsky, An abstract Brown–Douglas–Fillmore 

absorption theorem, Pacific J. Math. 198 (2001) 385–409], and establish sufficient conditions for Fredholm 

triples to be absorbing in a suitable sense. As an application of one of our criteria, we prove a multivariable 

Brown–Douglas–Fillmore type theorem. 


Keywords: K-Theory; C ∗ -Algebras 

1. Introduction and background 

We study the problem of simplifying the standard equivalence relation on KK-theory. The 

theorems obtained are applicable to a class of algebra that includes many real rank zero algebras, 

purely infinite algebras (simple or not), and many type I C ∗ -algebras. In hopes of making this 

paper somewhat self-contained, we shall include background material on KK-theory [10,11]. 

Since we make fundamental use of a purely large condition that was originally phrased in terms 

of the generalized Brown–Douglas–Fillmore group of extensions, Ext(A, B), we shall initially 

work in this setting; however, we later consider the case of Fredholm triples, in particular the class 

we term absorbing triples. The paper is organized as follows. In the first two sections we give 

definitions and establish some equivalent forms for the purely large property. In the third section, 

we give lemmas and technical results of various sorts. In Section 5 we establish a topological 

✩ Supported by NSERC, under grant #228065-00. 

E-mail address: dkucerov@unb.ca. 


doi:10.1016/j.jfa.2005.11.018

396 D. Kucerovsky / Journal of Functional Analysis 236 (2006) 395–408 

formulation of an absorption criterion that leads to a multivariable Brown–Douglas–Fillmore 

theorem in Section 6. In Section 7 we apply the techniques we have built up to the case of 

Fredholm triples, giving a nice sufficient condition for a triple to be absorbing. 

2. Preliminaries 

As pointed out by Kasparov [11], the Ext group can be defined as equivalence classes of absorbing 

extensions under unitary equivalence by multiplier unitaries, but on the other hand, it 

is often convenient to work with lifted Busby maps [5]. Thus, we define an extension to be an 

injective completely positive map into the multipliers which becomes a homomorphism when 

composed with the canonical map into the corona. The equivalence relation we consider is unitary 

equivalence by multiplier unitaries modulo the ideal. In order to effectively apply this model 

of Kasparov’s group KK 1 to concrete problems, one needs to be able to decide which extensions 

are absorbing. Before giving a relevant criterion, for the reader’s convenience we recall several 

definitions: 

Definition. Let B be a σ -unital, nuclear, and stable C ∗ -algebra. Let A be a separable C ∗ -algebra. 

(i) An extension τ : A → M(B) is said to be full if π ◦ τ : A → M(B)/B intersects no nontrivial 

ideal of the corona. Thus, π(τ(C ∗ (a))) ∩ I ={0} for all proper ideals I of the corona, 

and all positive a ∈ A. 

(ii) An extension τ : A → M(B) is said to be trivial if it is a homomorphism (rather than just a 

completely positive map). 

(iii) An extension τ : A → M(B) is said to be essential if the map π ◦ τ has no kernel. 

(iv) An extension τ : A → M(B) is said to be unital if A is unital and τ maps the unit of A to 

the unit of the multipliers. Otherwise, the extension is said to be nonunital. 

(v) An extension τ is weakly nuclear if the maps a ↦→ bτ(a)b ∗ are nuclear for every b in B.For 

an extension to be weakly nuclear, it is sufficient that either A or B is a nuclear C ∗ -algebra. 

(vi) The BDF sum [4] of two extensions τ and φ is the extension v1τv ∗ 1 + v2φv ∗ 2 where v1 and 

v2 are the generators of some given copy of O2 in the multipliers (such a copy exists if B is 

stable, and is unique up to unitary equivalence). 

(vii) A weakly nuclear extension π is said to be absorbing if the BDF sum of π with a weakly 

nuclear trivial extension φ is approximately unitarily equivalent to π, whenever φ and π are 

either both unital, or both nonunital. 

We notice that fullness, as defined above, is an extremely strong condition. However, unital 

extensions τ : A → M(B) of simple (unital) C ∗ -algebras A are necessarily full, as are absorbing 

extensions. 

3. C ∗ -algebraic absorption criteria 

Recalling [5] that an extension τ : A → M(B) can equivalently be given as a (semisplit) 

short exact sequence 0 → B → C → A → 0, we say that an extension is purely large if the 

extension algebra C has the property that, for every positive element c ∈ C + , either c is in B, or 

else the hereditary subalgebra cBc contains a stable subalgebra that generates B as an ideal. This 

definition is motivated by the following theorem [6], which is, in fact, the first of our C ∗ -algebraic 

absorption criteria. The importance of the theorem is that it shows (in the spirit of index theory!)

D. Kucerovsky / Journal of Functional Analysis 236 (2006) 395–408 397 

that a purely topological property, namely absorption, is equivalent to an algebraic property. The 

specific algebraic property in question is the somewhat technical-sounding purely large property 

that we have just defined, but we will shortly come up with nicer criteria. 

Theorem 3.1. Let A and B be separable C ∗ -algebras, with B stable. Let 0 → B → C → A → 0 

be an essential (unital) extension with weakly nuclear splitting map (a completely positive map 

s : A → C such that a ↦→ bs(a)b ∗ is nuclear for each b ∈ B). Then the following are equivalent: 

(i) The extension absorbs all trivial weakly nuclear (unital) extensions. 

(ii) The extension is purely large. 

(iii) The extension algebra has the approximation property that, for every c ∈ C + that is not zero 

in C/B, and every positive b in B, there is an element r ∈ B making the norm of b − rcr ∗ 

arbitrarily small. Moreover, the element r can be assumed to be in the unit ball if b and c/B 

are of norm one. 

It is a corollary of the above theorem that an extension τ : A → M(B) is absorbing if and 

only if its restrictions to subalgebras C ∗ (a), with a ∈ A + , are. In the main application of the 

theorem, it is customary to assume that the absorbing extension and the absorbed extension are 

unital or nonunital together. This stems from the observation that the BDF sum of a unital and a 

nonunital extension is never unital, so that a unital extension therefore cannot absorb a nonunital 

extension. It is not hard to check that, although this is not important in BDF theory, a nonunital 

absorbing extension in fact absorbs a unital extension, and it is not even necessary for the 

absorbed extension to be essential (the absorbing extension must, however, be essential). 

Definition. We say that an algebra B is an absorbing algebra if every full weakly nuclear extension 

of B by a separable algebra A is absorbing. 

Of course, K and O2 ⊗ K are the canonical examples of absorbing algebras [6]. In addition, 

many real rank zero algebras, all purely infinite stable algebras (simple or not) [21], and many 

type I C ∗ -algebras [17] are absorbing. 

We now prove a pleasantly geometrical if and only if criterion for absorption, phrased in terms 

of a weak form of Fredholmness for full projections. We also obtain several alternative forms of 

the criterion. 

Theorem 3.2. Let B be a stable and separable C ∗ -algebra. Then the following are equivalent: 

(i) B is an absorbing algebra. 

(ii) All full multiplier projections in M(B) have quasi-invertible image in the corona. 

(iii) All full multiplier projections are properly infinite. 

(iv) All full corona projections are quasi-invertible. 

(v) All full corona projections are properly infinite. 

(vi) All full multiplier projections majorize, in the corona, a corona projection equivalent to 1. 

In the statement of the above theorem, a projection is said to be full if the ideal it generates is 

the whole algebra. A projection P is said to be quasi-invertible if there exists an element x such 

that xPx ∗ = 1.


The above theorem will be further generalized in a future publication, [16], where we also 

apply it to the study of Rørdam’s group KL 1 (A, B). We notice that the theorem suggests that 

absorption can be characterized solely in terms of the image in the corona of an extension—this 

was, however, already pointed out in [6]. 

The proof is deferred to page 400, as we need to establish a few lemmas and propositions first. 

Remark 3.3. Since the canonical ideal is stable, quasi-invertibility in the multipliers is equivalent 

to quasi-invertibility in the corona. To see this, suppose that for some corona element c,wehave 

1 = rcr ′ for some r. Then, lifting c to any full element ˜c in the multipliers, we have ˜r ˜c˜r ′ = 1 + b 

for some element b of the ideal. But since the ideal is stable, we can find an isometry v such that 

v ∗ bv is small in norm, implying that v ∗ ˜r ˜c˜r ′ v is invertible. In fact, by the same type of argument, 

if an extension is full in the sense of Definition 2, then elements of the extension algebra that are 

not in the canonical ideal are in fact full in the multipliers. 

Remark 3.4. It follows from the theorem that an algebra B is absorbing if and only if every 

nonunital full extension of C by B is absorbing. 

We point out that an early version of part (ii) of this theorem motivated the corona factorization 

condition proposed in a preprint [13], and several conference talks, as part of a strategy for 

studying ideal-related absorption: 

Definition. A stable σ -unital C ∗ -algebra B has the corona factorization property if, for every 

positive c in M(B) + , either of the following equivalent conditions hold: 

(i) there is an r in M(BcB)/BcB such that rcr ∗ = 1inM(BcB)/BcB, and 

(ii) there is an r in M(BcB) such that rcr ∗ = 1inM(BcB). 

(The equivalence of (i) and (ii) follows from the fact that ideals in a stable C ∗ -algebra are 

stable, allowing a cut-down by a suitable isometry as in the above remark.) 

The corona factorization condition refined by means of a spectral condition is used in [14] to 

study ideal-related absorption. 

4. Lemmas and technical results 

We of course need to use several properties of purely large algebras in proving Theorem 3.2. 

The most important is our abstract Weyl–von Neumann type theorem: 

Lemma 4.1. [6] Let B be a separable stable C ∗ -algebra. Let C be a separable subalgebra of 

M(B), containing B and 1M(B) and having the purely large property with respect to B. Let 

φ : C → M(B) be a completely positive weakly nuclear map which is zero on B. Then there 

exists a sequence (vn) of isometries such that, for each c ∈ C, the expression φ(c) − v ∗ n cvn is 

in B, and goes to zero in norm as n →∞. 

Remark 4.2. In view of the condition on units, it is useful to note that the unitization of a purely 

large algebra is purely large. More precisely, if the image of C in the corona is not already unital, 

then one of the arguments in [6] shows that C + 1M(B) is purely large. The idea is the following. 

Given that for all positive c ∈ C − B, the hereditary subalgebra cBc contains a stable full (in B)


subalgebra, we are to show the same for (1 + c)B(1 + c). Because the image of C in the corona is 

not unital, there exists c ′ ∈ C such that c ′ (1 + c) is in C but not in B, so that (1 + c)c ′ Bc ′ (1 + c) 

contains a full stable subalgebra. But this then gives a full stable subalgebra of the larger algebra 

(1 + c)B(1 + c). This, in fact, is exactly how the nonunital version of the main result of [6] is 

derived from the easier unital case. 

The second major theorem that we need is the Kasparov stabilization theorem [10], one of 

the fundamental tools of Kasparov’s KK-theory [11], which will here be used to “diagonalize” a 

singly generated hereditary subalgebra of a stable algebra. One form of the Kasparov stabilization 

theorem is as follows. 

Theorem 4.3. [19] Let E be a Hilbert B-module that is countably generated in M(E). Ifwe 

denote the standard Hilbert B-module by HB, then E ⊕ HB is unitarily equivalent to HB. 

In the statement of the theorem, a Hilbert module E is said to be countably generated in 

M(E) if there is a sequence of multipliers (mi) ⊆ M(E) such that the elements {mib: b ∈ B} 

span a dense submodule of E. This definition generalizes Kasparov’s, since the generators mi do 

not need to be in the module E. It is likely that the assumption of σ -uniticity can be removed in 

the next theorem. 

Theorem 4.4. Let B be stable and σ -unital. A hereditary subalgebra, ℓBℓ ∗ , generated by an 

element ℓ of the multipliers M(B) is isomorphic to a hereditary subalgebra generated by a 

multiplier projection, P .Ifℓ is not contained in a proper ideal of the multipliers, then neither 

is P . 

Proof. The closed right ideal E := ℓB is, if we take B to act in the natural way from the right, 

a Hilbert B-module with inner product 〈a,b〉:=a∗b, and is countably generated (in the generalized 

sense described above) with generators (ℓ1/n ) ∞ n=1 . Thus, by the above form of the Kasparov 

stabilization theorem (Theorem 4.3), there is a unitary U in L(E ⊕ HB, HB) implementing an 

isomorphism of E ⊕ HB and HB. 

Let P be the projection of E ⊕ HB onto the first factor, E. The projection T := UPU∗ ∈ 

L(HB) has image isomorphic (by a unitary equivalence) to E, and thus by the definition of the 

compact operators on a Hilbert module, 

K(T HB) ∼ = K(ℓB). 

Now, however, recalling the definition of the compact operators on a Hilbert module, we see that 

K(ℓB) is generated by elements of the form ℓb1b ∗ 2 ℓ∗ , where the bi are in B. Hence (up to unitary 

equivalence), T K(HB)T ∼ = ℓBℓ ∗ . However, T is in L(HB) = M(B ⊗K), and K(HB) = B ⊗K, 

which thus completes the proof of the first part of the lemma since B is stable. For the fullness 

claimed in the last part of the theorem, notice that if we write L(E ⊕ HB, HB)P L(HB,E⊕ HB) 

as a formal two-by-two matrix, and consider the lower right-hand corner, we are to show that 

L(ℓB, HB)L(HB,ℓB)is dense in L(HB). However, 

L(ℓB, HB)L(HB,ℓB)= L(B, HB)ℓ ∗ ℓL(HB,B) 

= L(B, HB)L(B)ℓ ∗ ℓL(B)L(HB,B)


and since ℓ ∗ ℓ is full in L(B) by hypothesis, the last expression simplifies to L(B, HB)L(HB,B), 

which is equal to L(HB) if B is stable and σ -unital. ✷ 

Corollary 4.5. If D is a hereditary σ -unital subalgebra of the σ -unital stable algebra B, then 

M(D) can be identified with a corner inside M(B). 

Last, we recall the definition of properly infinite positive elements [12]. 

Definition. A positive element P in a C∗-algebra C is said to be properly infinite if P ⊕ P P 

in M2(C), where a b means that rnbr∗ n → a in norm for some sequence (rn). 

The main fact that we need about such elements is the following well-known result, due to 

J. Cuntz in the simple case, with generalizations by M. Rørdam (and possibly others). 

Lemma 4.6. A full properly infinite projection P has the property that xPx ∗ = 1 for some element 

x. 

The lemma is established by noticing that if rnbr∗ n → 1 then rnbr∗ n is invertible for some 

sufficiently large n. 

Proof of Theorem 3.2. We now show that (i) implies (ii). Thus, we assume that all weakly 

nuclear full extensions are absorbing, and prove quasi-invertibility. If P is a full multiplier projection, 

then if π(P) is equal to 1 in the corona, we can readily show, using Remark 3.3, that 

P is quasi-invertible in the multipliers. We now have the case where P is not equal to 1 in 

the corona. The algebra C ∗ (1,P,B) can be regarded as the extension algebra of some trivial 

full extension. Since this extension is absorbing by hypothesis (and is weakly nuclear because 

the quotient algebra is abelian, hence nuclear), the algebra C := C ∗ (1,P,B) is purely large. 

If Q = 1 is a multiplier projection equivalent to 1M(B), there is an obvious homomorphism 

δ : C ∗ (1, π(P )) → C ∗ (1,Q). We can thus regard δ as a homomorphism from C ∗ (1, π(P )) into 

the multipliers. The unital map δ ◦π : C → M(B) satisfies the hypotheses of Lemma 4.1, so there 

are isometries (vn) such that δ ◦ π(c) − v ∗ n cvn is in the canonical ideal B and, moreover, goes 

to zero pointwise as n →∞. In particular, Q − v ∗ n Pvn can be made arbitrarily small in norm. 

Since Q = WW ∗ for some multiplier isometry W , it follows that 1 − W ∗ v ∗ n PvnW is small in 

norm, implying that W ∗ v ∗ n PvnW is invertible. This finishes the proof that (i) implies (ii). 

We now show that (ii) implies (i). 

By Theorem 3.1 it is sufficient to show that cBc contains a stable full subalgebra for all positive 

full multiplier elements c ∈ M(B). By Theorem 4.4, it is sufficient to show this for the case 

where c is a full projection in the multipliers. By hypothesis, the projection c is quasi-invertible, 

so that 1 = x ∗ cx, implying that cx is an isometry. But this isometry gives a homomorphism 

Ad(cx) : B → cBc, and the image of the homomorphism is the desired stable full subalgebra. 

The equivalence with the other conditions listed is straighforward, using Lemma 4.6 and Remark 

3.3. ✷ 

5. An enveloping algebra criterion for absorption 

In this section we give a completely different set of absorption criteria, applicable to single 

extensions rather than to the set of all full extensions. We shall find that every positive element of


the image of an absorbing extension is contained, up to unitary equivalence, in a canonical zerodimensional 

abelian subalgebra that we denote W . This has highly interesting consequences for 

the functional calculus, leading us to a multivariable functional calculus problem that we solve 

by means of a multivariable BDF-type theorem. 

We shall need the following topological result, characterizing the Stone–Čech corona of the 

positive integers. 

Theorem 5.1. (Parovičenko [18,22]) A totally disconnected compact F-space without isolated 

points, having weight ℵ1, and such that every zero-set is a regular closed set, is homeomorphic 

to β(N)/N. Moreover, β(N)/N maps onto every compact space that has weight at most ℵ1. 

In this section, we specialize to stably unital separable algebras. By Brown’s theorem [3], 

a separable algebra is stably unital if and only if the stabilization contains a full projection. We 

may as well, therefore, reduce to the case of B ⊗ K where B is a unital and separable C ∗ -algebra. 

Define a map ¯δ : ℓ ∞ → M(B ⊗ K) by embedding ℓ ∞ in the natural way into 1 ⊗ M(K) ⊂ 

M(B ⊗ K). Clearly this is a homomorphism, and composing with the natural quotient map 

π : M(B ⊗ K) → M(B ⊗ K)/B ⊗ K, we find that the kernel of π ◦ ¯δ is exactly the space of 

sequences converging to zero. Thus we have an injective homomorphism, denoted δ, from ℓ ∞ /c0 

to M(B ⊗ K)/B ⊗ K. 

Noticing that c0 can be thought of as C0(N), where the integers N have the usual discrete 

topology, we see that the space of characters of ℓ ∞ /c0 ∼ = M(C0(N))/C0(N) is β(N) \ N, where 

β(N) is the Stone–Čech compactification of the natural numbers. Hence, by the Gelfand theorem, 

δ can be identified with a map from the highly nonseparable algebra C(β(N) \ N) into M(B ⊗ 

K)/B ⊗ K. The map ¯δ maps C(β(N)) into M(B). 

Let us henceforth denote the image of ¯δ by ¯W , and the image in the corona by W . Note, 

however, that we do not obtain an injective map of β(N) \ N into the multipliers. From the 

viewpoint of extension theory, we have an apparently nontrivial Busby map τ : β(N) \ N → 

M(B)/B. If we want an essential extension, there does not appear to be any way to make this 

extension trivial. (It is, in principle, possible that there is some other construction that does give 

a trivial essential extension τ : β(N) \ N → M(B)/B.) 

We point out that the usual lifting theorems that convert a Busby map into a completely 

positive map into the multipliers all require separability of the source algebra. Therefore, it is 

difficult to show that we can represent τ by a completely positive map into the multipliers unless 

we restrict to a separable subalgebra of β(N) \ N. We return to the issue of problems arising 

from nonseparability later. The next proposition studies the process of restriction to separable 

subalgebras. First a lemma (which is actually [7, Exercise 3.2.I]). 

Lemma 5.2. An unital abelian C ∗ -algebra has a countable dense subset if and only if the Gelfand 

spectrum has a countable base as a topological space. The algebra has a dense subset of cardinality 

at most ℵ1 if and only if the Gelfand spectrum has a base of cardinality at most ℵ1. 

Proof. Suppose that we are given a dense subset D of C(X), where X is compact and Hausdorff. 

We are to show that there is a base of X of cardinality |D|. We claim that the sets B(f ) := {x ∈ X: 

|f(x)| < 1}, with f ∈ D, form such a base. Given an open set O, choose a point x ∈ O. Since 

X is completely regular [8], there is a function s : X →[0, 1] that is 0 at x and is 1 outside O. 

Approximate 2s within ɛ = 1/4byafunctionf ∈ D. It follows that B(f ) contains x and is itself 

contained in O. It is now clear that the B(f ) do form a base (of cardinality |D|).


For the converse, use complete regularity to find a base whose elements are co-zero sets (a cozero 

set is {x: f(x)>0} for some function f ) by shrinking the elements of the given base. This 

may lower the cardinality of the base, but cannot increase it. Thus, we have a family of functions 

supported on basis elements. The finite linear combinations with rational coefficients, and their 

finite products, form an algebra that is (by the Stone–Weierstrass theorem) dense in C(X). Since 

we only allow multiplication by rationals plus the formation of finite products and sums, the 

cardinality of this algebra is bounded by max{ℵ0, |B|}, where B is the base we started with. The 

bound on cardinality is obtained by use of Cantor’s result that |A × B| max{|A|, |B|} if one of 

the sets A or B is infinite. ✷ 

In the next proposition, the restriction to injective Busby maps is necessary. Without injectivity, 

counter-examples can be found. 

Proposition 5.3. Given a unital abelian algebra A with a dense subset of cardinality at most ℵ1, 

there is a commutative diagram 

C(β(N)) ∼ = ℓ ∞ (N) 

0 

 

B ⊗ K 

 

¯δ 

−→ M(B ⊗ K) → 0 

 

 

A → C(β(N) \ N) ∼ = ℓ∞ δ 

(N)/c0(N) −→ M(B ⊗ K)/B ⊗ K → 0 

The (Busby) map from A to M(B ⊗ K)/B ⊗ K defined by this diagram is injective, purely large, 

and nuclear. It is trivial if and only if the Gelfand spectrum of A has a countable dense subset. 

Proof. By the Gelfand theorem and the previous proposition A = C(X) for some compact Hausdorff 

space X with a topological basis of cardinality ℵ1 or less. 

First, suppose the extension is trivial. There then is an injective unital homomorphism from A 

into the image of ¯δ in M(B ⊗K). Composing this injection with ¯δ −1 , we obtain a unital injection 

of A into C(β(N)), which by the Gelfand theorem corresponds to a mapping g of β(N) onto X. 

The inverse image g −1 (V ) of an open set V of X must intersect N since N is dense in β(N), 

showing that g(N) is a countable dense subset of X. 

Conversely, if we assume that X has a countable dense subset, this countable dense subset 

gives us a continuous mapping h : N → X with dense image in X. Using the fundamental property 

of Stone–Čech compactifications to extend h uniquely to a map β(h): β(N) → X, wesee 

that the image of β(h) is compact, hence closed, and dense in X. Therefore the image is all of X, 

and the pullback of β(h) is thus an injective homomorphism of A into C(β(N)), which we then 

compose with ¯δ to obtain (the lifting of) a trivial Busby map. (The map β(h) is unique up to 

homeomorphism of β(N).) 

This completes the proof for the separable (or trivial) case. We now consider the nonseparable 

(or nontrivial) case. By the lemma and by Parovičenko’s theorem (Theorem 5.1), the space 

β(N)/N maps onto X, so we have a unital injection of A into C(β(N) \ N). (This map is in fact 

unique up to homeomorphism of β(N) \ N.) Composing with the map δ we have a injective map 

π


τ into M(B)/B. Since A is commutative, the map is nuclear. Recall that the extension algebra 

C associated with the constructed Busby map, τ , is by definition the pullback 

C −→ A 

τ 

 

 

π 

M(B ⊗ K) −→ M(B ⊗ K)/B ⊗ K 

Since τ has no kernel, we may as well regard C as being identified with its image in M(B ⊗ K), 

and then by comparing with the above diagram, we see that the extension algebra is in fact a 

subalgebra of the image ¯W of ¯δ in M(B ⊗ K). 

We notice that the image algebra W is purely large: choosing some positive element (λi) of 

ℓ ∞ that is not in c0,themap¯δ gives an element of 1⊗B(H) that majorizes some nonzero multiple 

of a projection P in the image of ¯δ. Since this projection can be taken to be nonzero on infinitely 

many λi, it is properly infinite. Thus, P is a properly infinite element of M(B ⊗ K), and there 

exists x ∈ M(B ⊗ K) such that xPx ∗ = 1M(B⊗K). Since the given element majorizes P up 

to a scalar multiple, we can therefore find y ∈ M(B ⊗ K) such that y ¯δ((λi))y ∗ = 1. Therefore 

the hereditary subalgebra ¯δ((λi))(B ⊗ K)¯δ((λi)) contains a subalgebra that is stable and full in 

B ⊗ K. Strictly speaking, we must also check the case of perturbation by elements of B ⊗ K, 

thus we should show a similar result for (¯δ((λi)) + b)(B ⊗ K)(¯δ((λi)) + b) where b ∈ B ⊗ K. 

But using stability we can easily find, as in Remark 3.3, a y ′ such that y ′ (¯δ((λi)) + b)y ′∗ = 

1M(B⊗K). ✷ 

We now arrive at a very interesting absorption criterion. The criterion will be stated in terms 

of “copies of W ,” a term with several possible interpretations, so let us take a few moments to 

discuss this before continuing. A copy of W is simply the extension algebra of some (generally 

nontrivial) injective and purely large Busby map ℓ ∞ /c0 → M(B ⊗ K)/B ⊗ K. These extensions 

have the peculiar property, described in Proposition 5.3, of having separable subalgebras coming 

from trivial extensions, without being themselves trivial. We would naturally expect them to 

be absorbing, however, we have difficulty even lifting them to completely positive maps into the 

multipliers, which would certainly be a necessary first step in showing this. Both the Choi–Effros 

and the Haagerup lifting theorems have separability of the source algebra as a hypothesis, and, 

moreover, the Elliott–Kucerovsky absorption theorem also requires separability. The work of 

Hadwin [9] suggests that fundamental differences between the separable and nonseparable cases 

are to be expected. As a result, we must leave open for the moment the quite interesting question 

of whether copies of W are unitarily equivalent by multiplier unitaries. In Theorem 6.1 we prove 

that this is true if we restrict to finitely generated subalgebras (within given copies of W ), and 

this is sufficient for most applications: speaking loosely, one could say that there is a canonical 

copy of W after restriction to separable subalgebras. 

Corollary 5.4. Let A be a separable unital C ∗ -algebra and B be the stabilization of a unital 

separable C ∗ -algebra. Let τ : A → M(B)/B be a trivial full extension. Then, τ is absorbing if 

and only if for each positive a ∈ A, the algebra τ(C ∗ (a)) is contained in a copy of W. 

Proof. It follows from the basic absorption criterion (Theorem 3.1) that if τ : A → M(B)/B is 

absorbing, then any restriction of τ to a subalgebra of A is absorbing. Thus, the restriction of τ to 

a subalgebra of the form C ∗ (a), with a being some positive nonzero element, is absorbing. Being


still trivial it is thus unitarily equivalent to the trivial extension obtained from Proposition 5.3, 

and we have proven the “only if” part of the theorem. 

We prove the converse direction next, but in greater generality. ✷ 

Theorem 5.5. Let A be a separable unital C ∗ -algebra and B be the stabilization of a unital 

separable C ∗ -algebra. Let τ : A → M(B)/B be a full extension, not necessarily trivial. 

Then, τ is absorbing if for each positive a ∈ A, the algebra τ(C ∗ (a)) is contained in a copy 

of W . In this case, the restricted extensions τ(C ∗ (a)) are necessarily trivial. 

Proof. Let us denote the restricted extensions by τa. It follows from Theorem 3.1 that τ is 

absorbing if and only if all the τa are. 

We first prove that each τa is, in fact, trivial. Choosing some particular τa, the image algebra 

D of τa is by hypothesis contained in W ∼ = ℓ ∞ /c0. Since the spectrum of a is some subset of 

[0, 1], the algebra D is singly generated. Letting g be the generator, we lift it to ¯g in ℓ ∞ .The 

spectrum of ¯g may be larger than that of g by some countable set, but we can find a function 

f : (0, 1]→(0, 1] such that f is the identity map when restricted to the spectrum of g but maps 

the spectrum of ¯g to the spectrum of g. Applying this function, we thus have a lifting, ¯g, that 

is isospectral to g, and then Gelfand’s theorem gives us a splitting of τa. Now,τa is trivial, 

with a splitting map s : D → M(B) whose image is in a copy of ¯W. The absorption property 

is a property of the image of an extension in the corona. Therefore, τa is absorbing because the 

image is contained in a copy of W , or more exactly, is contained in the image of a copy of the 

extension of Proposition 5.3. ✷ 

We remark that it is quite possible for the restrictions of an extension, as in the above theorem, 

to be trivial without the whole extension being trivial. In such a case, we have local splitting 

homomorphisms that do not assemble to give a globally defined splitting homomorphism. 

Finally, one may inquire if the triviality condition can be dropped from Corollary 5.4. 

Corollary 5.6. Let A be a separable unital C ∗ -algebra and B be the stabilization of a unital separable 

bootstrap class C ∗ -algebra with K1(B) ={0}. Let τ : A → M(B)/B be a full extension. 

Then, τ is absorbing if and only if for each positive a ∈ A, the algebra τ(C ∗ (a)) is contained in 

a copy of W. 

Proof. Notice the spectrum of a is a closed subset of [0, 1]. It follows that K1(C ∗ (a)) ={0}, 

and that K0(C ∗ (a)) is a free abelian group. Thus, by the UCT and the properties of countably 

generated abelian groups, we find that KK 1 (C ∗ (a), B) is zero, and therefore, the restrictions of 

τ are—if absorbing—always trivial extensions. We now have the claimed result by combining 

the two previous results. ✷ 

In summary, we have shown that: 

(i) For trivial extensions τ , absorption is equivalent to τ(C ∗ (a)) being contained (up to unitary 

equivalence) in W . 

(ii) For not necessarily trivial extensions, the above condition is still sufficient for absorption, 

but may not be necessary unless the K1-group of the canonical ideal is zero.


It would, of course, be possible to replace the K-theoretical condition on the algebra, 

K1(B) ={0}, by a less restrictive but more technical KK-theoretic condition on τ . 

6. A multivariable Brown–Douglas–Fillmore theorem 

The absorption criteria of the previous section may seem at first glance to be of largely theoretical 

interest, but in fact they can be used to construct an extended functional calculus for 

suitable elements of the corona, by using properties of zero-dimensional topological spaces to 

find elements of the enveloping algebra W having interesting properties. This extended functional 

calculus has been partially explored in [15]. As a consequence, the following question arises naturally: 

given (commuting) elements d1,...,dn from one copy of W , and elements v1,...,vn 

from another copy of W, when is C ∗ (d1,...,dn) unitarily equivalent to C ∗ (v1,...,vn)? 

We now proceed to resolve this question, which may be thought of in terms of the multivariable 

functional calculus: characterize unitary equivalence classes of n-tuples from two different 

abelian subalgebras. We note that in some cases, W may indeed be a masa. 

Thus, suppose that we are given two such n-tuples, (d1,...,dn) and (v1,...,vn). The algebra 

D1 generated by the di is, by Gelfand’s theorem, isomorphic to some C(X) but is not usually 

singly generated. Since D1 is contained in some copy of W , and the Gelfand spectrum of W 

is totally disconnected, we can approximate each di by a finite sum of projections from W . 

Approximating di by an element of this form, we have sin with sin − di < 1/n. Thesetof 

projections (pjin) used to form the sin is a countable family, and the algebra D1 lies in the 

algebra generated by these projections. Relabling the countable family as (qm), we notice that by 

the Stone–Weierstrass theorem, the element 

g := 

∞ 

1 

2qm − 1 

3 m 

generates the algebra C ∗ (pjin). (This argument is based on an idea of von Neumann’s. Compare 

[20] and [1].) Denoting the generator by g, we notice that it has countable spectrum. Thus, we 

may lift to ¯g in ¯W and apply a function to ¯g to insure isospectrality with g. It follows that 

there is, by Gelfand’s theorem, a splitting map s from C ∗ (pjin) to ¯W ⊂ M(B). Restricting the 

splitting map to D1, we see that there is a trivial (absorbing) extension τ : C(X) → M(B)/B 

with image exactly D1, and similarly for the D2 generated by the vi. We now have a theorem of 

Brown–Douglas–Fillmore type, since two trivial absorbing extensions are unitarily equivalent if 

and only if the image algebras are isomorphic. 

Theorem 6.1. If (ui) and (di) are two tuples of elements of two distinct copies of W , then the 

two abelian C ∗ -subalgebras of the corona, C ∗ (u1,...,un) and C ∗ (d1,...,dm), are unitarily 

equivalent by a multiplier unitary if and only if the Gelfand spectra of the two algebras are 

isomorphic. 

7. Fredholm triples 

For the reader’s convenience we first summarize the basic facts about the Fredholm module 

picture of KK-theory. Since there are many equivalent ways to define KK 1 (A, B) in terms of 

Fredholm modules, and no one canonical choice, we follow the definition in [2, Section 17.6.4], 

involving approximate (ungraded) projections.


Definition. Let B be a σ -unital stable C ∗ -algebra, and let A be a separable C ∗ -algebra. A stabilized 

ungraded Fredholm cycle in KK 1 (A, B) is a triple (M(B), φ, F ) where φ is a homomorphism 

from A to M(B), and F ∈ M(B) is such that: 

(i) (F ∗ F − F)φ(a)∈ B; 

(ii) [F,φ(a)]∈B; and 

(iii) φ(a)(F − F ∗ ) ∈ B. 

A cycle is said to be trivial if the above three expressions are actually zero. 

We can summarize the above definition by saying that Fredholm triples are specified by a homomorphism 

φ and an operator that is an approximate projection: more specifically, an element 

of Iφ that becomes a projection in Iφ/Jφ, where 

Iφ := m ∈ M(B): φ(a),m ∈ B for all a ∈ A , 

Jφ := m ∈ M(B): φ(a)m∈ B for all a ∈ A . 

Sometimes, Fredholm triples in KK 1 (A, B) are specified by self-adjoint approximate unitaries 

instead, but this is equivalent since self-adjoint unitaries map to projections under the map 

u ↦→ 1 + u/2. For more information on the various pictures of KK-theory, and the interesting 

transformations that relate one to the other, see [2]. 

In some applications, cycles arise which are not stabilized, meaning that M(B) is replaced 

by the C ∗ -algebra of adjointable operators on some given Hilbert B-module, but it can be shown 

that under an appropriate definition of equivalence unstabilized cycles are nevertheless always 

equivalent to stabilized cycles. 

There are a number of apparently quite different equivalence relations on KK 1 -cycles, and 

the one most relevant to our situation is given by “compact” perturbation, addition of degenerate 

cycles, and unitary equivalence by multiplier unitaries. (By “compact” perturbation is meant 

perturbation of the operator F by elements of Jφ.) 

Definition. We say that a Fredholm triple (M(B), φ, F ) is unital if and only φ(1) = 1 + b and 

F = 1 + b ′ for some b,b ′ ∈ B. 

Clearly this is an extremely strong property. It is, however, of interest in that it plays a role in 

the definition of the absorption property for a Fredholm triple. 

Definition. A unital Fredholm triple F := (M(B), φ, F ) is absorbing if F ⊕ T is unitarily 

equivalent to F for all unital trivial cycles T . A nonunital Fredholm cycle F is absorbing if it 

has this property for all trivial cycles T . 

One observes that there is a natural isomorphism 

KK 1 (A, B) → Ext(A, B), 

(φ, F ) ↦→ π(FφF ∗ )


and that the equivalence relation on Ext exactly corresponds to the stabilized cp equivalence 

relation in KK 1 , meaning equivalence modulo perturbation by Jφ, unitary equivalence, and addition 

of stabilized degenerate cycles [2, Section 17.6.4] which may be taken to be unital if the 

original cycle is unital [11]. Addition of stabilized degenerate cycles corresponds to the addition 

of trivial extensions in Ext, so we see that for absorbing cycles (φ1,F1) and (φ2,F2), wehave 

that (φ1,F1) ∼cp (φ2,F2) in KK 1 if and only if π(F1φ1F ∗ 1 ) ∼u π(F2φ2F ∗ 2 ). 

We now give an example showing that a cycle can be absorbing even if (1 − F)φ(·)(1 − F) 

is small. 

Example 7.1. Let φ be the Kasparov GNS representation of A on B: thus, φ : A → M(B ⊗ K) 

is given by 1 ⊗ π where π is the usual universal representation of A on B(H) ∼ = M(K). Then it 

follows from Kasparov’s results [11] that (M(B), φ, 1) ∈ KK 1 (A, B) is an absorbing triple. The 

complement (M(B), φ, 0) is zero. We can elaborate this example somewhat, by letting F = e11 

in M2(M(B ⊗ K)), and then letting φ be the direct sum of Kasparov’s GNS representation in 

the e11 corner and any arbitrary homomorphism in the e22 corner. 

It would be desirable, of course, to give elegant necessary and sufficient condition for a 

specific triple to be absorbing. This is a hard problem, however, we point out one very pretty 

sufficient condition. An operator F ∈ M(B) is said to be quasi-Fredholm if it is quasi-invertible 

in the corona, or equivalently, if there are x,y ∈ M(B) such that xFy = 1 + b for some b ∈ B. 

Theorem 7.2. Let B be stable and separable. A Fredholm triple (M(B), φ, F ) ∈ KK 1 (A, B) is 

absorbing if F φ(a)F ∗ is a quasi-Fredholm operator whenever it is positive. 

Proof. Let C be the extension algebra of the associated Busby map a ↦→ F φ(a)F ∗ . A positive 

element of C not in B is of the form F φ(a)F ∗ + b for some a ∈ A and b ∈ B, where F is the 

given essential projection from the Fredholm triple. 

The hypothesis implies that there is r1 and r2 in the corona such that d := F φ(a)F ∗ satisfies 

r1dr2 = 1Q. But then, 1 r1d 2 r ∗ 1 r2r ∗ 2 , so that there is an r′ with r ′ d 2 r ′∗ = 1. Lifting r ′ to 

˜r ′ in the multipliers, we have that the original element d + b satisfies ˜r ′ (d + b) 2 ˜r ′∗ = 1 + b0 

for some b0 in the canonical ideal, and we can cut down by a isometry obtained from stability, 

obtaining v ∗ ˜r ′ (d + b) 2 ˜r ′∗ v = 1 + v ∗ b0v with the norm of v ∗ b0v small. Thus, this expression is 

invertible, and r ′′ (d + b) 2 r ′′∗ = 1forsomer ′′ in the multipliers. We now have an isometry V := 

(d + b)r ′′∗ that implements an equivalence of VBV ∗ ⊂ (d + b)B(d + b) and B, showing that 

(d + b)B(d + b) contains a stable (full) hereditary subalgebra. Thus, the absorption criterion of 

Theorem 3.1 holds. Actually, the conclusion of that theorem implicitly has two cases, according 

to whether the given extension is unital or not, however, in both cases the hypothesis just involves 

existence of stable full hereditary subalgebras as shown above. ✷ 

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Second order initial boundary-value problems of 

variational type 

Denis Serre 

École Normale Supérieure de Lyon, France 1 

Received 29 August 2005; accepted 26 February 2006 


Communicated by H. Brezis 

Abstract 

We consider linear hyperbolic boundary-value problems for second order systems, which can be written 

in the variational form δL = 0, with 

 

|∂t 

L[u]:= u| 2 − W(x;∇xu) dx dt, 

F ↦→ W(x; F)being a quadratic form over Md×n(R). The domain of L is the homogeneous Sobolev space 

H˙ 1 (Ω × Rt ) n , with Ω either a bounded domain or a half-space of Rd . The boundary condition inherent 

to this problem is of Neumann type. Such problems arise for instance in linearized elasticity. When Ω is 

a half-space and W depends only on F , we show that the strong well-posedness occurs if, and only if, the 

stored energy 

 

W(∇xu) dx 

Ω 

is convex and coercive over ˙ 

H 1 (Ω) n . Here, the energy density W does not need to be convex but only 

strictly rank-one convex. The “only if” part is the new result. A remarkable fact is that the classical characterization 

of well-posedness by the Lopatinskiĭ condition needs only to be satisfied at real frequency pairs 

(τ, η) with τ 0, instead of pairs with ℜτ 0. Even stronger is the fact that we need only to examine pairs 

(τ = 0,η), and prove that some Hermitian matrix H(η)is positive definite. Another significant result is that 

E-mail address: denis.serre@umpa.ens-lyon.fr. 

1 UMPA (UMR 5669 CNRS), ENS de Lyon, 46, allée d’Italie, F-69364 Lyon, cedex 07, France. The research of the 

author was partially supported by the European IHP project “HYKE”, contract # HPRN-CT-2002-00282. 


doi:10.1016/j.jfa.2006.02.020

410 D. Serre / Journal of Functional Analysis 236 (2006) 409–446 

every such well-posed problem admits a pair of surface waves at every frequency η = 0. These waves often 

have finite energy, like the Rayleigh waves in elasticity. When we vary the density W so as to reach nonconvex 

stored energies, this pair bifurcates to yield a Hadamard instability. This instability may occur for 

some energy densities that are quasi-convex, contrary to the case of the pure Cauchy problem, as shown in 

several examples. At the bifurcation, the corresponding stationary boundary-value problem enters the class 

of ill-posed problems in the sense of Agmon, Douglis and Nirenberg. For bounded domains and variable 

coefficients, we show that the strong well-posedness is equivalent to a Korn-like inequality for the stored 

energy. 


Keywords: Initial boundary-value problem; Lopatinskiĭ condition; Convex energy 


Let W be a quadratic form on the space Md×n(R) of d × n matrices with real entries. For 

the sake of simplicity, we assume in this introduction that the spatial domain is a half-space: 

Ω := Rd−1 × (0, +∞). Bounded domains will be considered in Section 7. We consider the 

stored energy functional 

 

W[u]:= W(∇xu) dx 

Ω 

for a field u : Ω → R n . In the context of Calculus of Variation, this energy is associated to a 

second order differential operator P : H 1 (Ω) n → H −1 (Ω) n defined by 

(Pu)j =− 

d 

α=1 

∂α 

∂W 

∂Fαj 

and to the Neumann-type boundary operator B, defined by 

(Bu)j := ∂W 

∂Fdj 

 

(∇xu) , j = 1,...,n, 

(∇xu), j = 1,...,n. 

Recall that when Pu ∈ L 2 (Ω) n , then Bu is in H −1/2 (∂Ω) n . 

We are concerned in this paper with the evolution boundary-value problem (IBVP) associated 

with the Lagrangian 

where 

 

L[u]:= 

R 

 

Ek[u]:= 

is the kinetic energy. Such an IBVP has the form 

Ek[u]−W[u] dt, 

Ω 

1 

2 |∂tu| 2 dx

D. Serre / Journal of Functional Analysis 236 (2006) 409–446 411 

∂ 2 t 

u + Pu = f (x ∈ Ω, t ∈ R), (1) 

Bu = g (x ∈ ∂Ω, t ∈ R) (2) 

with P and B as above. 

In practical situations, W is not quadratic and Ω is a general domain, say a bounded one. 

However, it is well known from the works by Kreiss [10], Sakamoto [15] and Majda [12] that the 

well-posedness of a general IBVP is intimately related to that of IBVPs with frozen coefficients 

in half-spaces, obtained by linearization about uniform states. This explains the relevance of 

problem (1), (2) where P and B are linear with constant coefficients and Ω is a half-space. Such 

a problem is a building block in the general theory. In general, passing from the constant coefficient 

case to the variable coefficient case requires much involved tools, like pseudo-differential 

estimates, symbolic symmetrizers and so on. This passage will be much simpler in the situation 

studied here, as shown in Section 7. 

We point out that the right-hand side f,g have a variational origin, if we add integrals 

 

R 

 

Ω 

f · udxdt and 

 

 

R ∂Ω 

g · udydt 

to the Lagrangian. As suggested by these expressions, we denote by x the space variable interior 

in Ω and by y the variable along the boundary ∂Ω. Therefore x = (y, xd). 

Throughout this paper, we make the minimal assumption that the wave-like operator ∂2 t + P 

is hyperbolic. 2 This amounts to saying that W is strictly rank-one convex, which means that for 

every F in Md×n(R), 

(rk F = 1) ⇒ W(F)>0 . (3) 

If W is convex, then it is obviously rank-one convex. Rank-one convexity amounts to saying 

that the stored energy W is convex over H˙ 1 (Rd ) n , when the domain is the whole space Rd instead of a half-space. Let us recall that W is polyconvex3 if it is the sum of a convex quadratic 

form W0 and of a form that vanishes on the cone of rank-one matrices. The latter forms are 

called (quadratic) null-Lagrangians or null-forms. Such null-forms do not contribute to the stored 

energy if Ω = Rd , but contribute through boundary integrals when Ω is a general domain. In 

other words, a null-form does not contribute to the operator P, but contributes to the boundary 

operator B. Quadratic null-forms are linear combinations of 2 × 2 minors of F . Polyconvexity 

obviously implies rank-one convexity, and the converse is true if and only if min{d,n} 2(see 

[16,18]). 

When the stored energy W is convex and the boundary condition is homogeneous (that is 

g ≡ 0), then the IBVP admits a convex total energy 

E = Ek + W. 

2 Some specialists say strongly hyperbolic. 

3 The definition of polyconvexity, due to J. Ball, is more elaborated for general (non-quadratic) functions W .


When W is coercive on ˙ 

H 1 (Ω), the IBVP has the abstract form 

dU 

dt + AU = ˜ 

f, U := (∂tu, ∇xu), 

where A is a maximal 4 monotone operator over L 2 (Ω) (d+1)n , this space being endowed with 

the norm induced by E. Therefore the homogeneous initial boundary-value problem (IBVP) is 

well-posed, in the sense that −A generates a continuous semi-group of contractions with respect 

to the norm E. Actually, because of conservativity, it generates a group of E-isometries (St)t∈R. 

Given a in the domain of A, t ↦→ Sta =: U(t) is the unique strong solution of U ′ + AU = 0 

such that U(0) = a. Then St is extended to L 2 by density, thanks to the contractivity. When f is 

non-zero, the homogeneous IBVP is solved through the Duhamel’s principle 

 

U(t)= StU(0) + 

This solution satisfies the well-known a priori estimate 

e −2γT ∇x,tu(T ) 2 

L 2 + γ 

C 

0 

T 

∇x,tu(0) 2 

L 2 + 1 

γ 

0 

t 

St−τ ˜ f(τ)dτ. 

e −2γt ∇x,tu(t) 2 

L 2 dt 

T 

0 

e −2γtf(t) 2 L2 

dt , (4) 

where C is a finite number, independent of either (u, f ), orγ>0 and T>0. We point out that 

this estimate shares the scaling invariance of the IBVP. 

Droping the convexity/coercivity assumption for W, we say that the homogeneous IBVP is 

strongly well-posed if it satisfies an estimate of the form (4). This is the same inequality than that 

considered by Kreiss or Sakamoto, except for the boundary terms, which must be absent in our 

homogeneous case. The main goal of this paper is to characterize these variational problems that 

are strongly well-posed. We show that they are precisely those for which the stored energy W is 

coercive over H˙ 1 (Ω). 

Of course, this does not mean that W be convex over Md×n(R). Therefore there is a need of 

a practical tool in order to characterize the densities W that yield strongly well-posed IBVPs. In 

the context of general hyperbolic IBVPs, the appropriate concept is that of Lopatinskiĭ condition. 

This is an algebraic property, which must be checked at every non-zero frequency pair (τ, η), 

with ℜτ 0(theuniform Lopatinskiĭ condition) and η ∈ Rd−1 . The situation turns out to be 

much simpler in our variational context: we show that it is sufficient to check the Lopatinskiĭ 

condition at real pairs (τ, η) ∈ Rd \{0, 0}. We find an even simpler characterization in terms of 

the simple modes of the stationary equation: if η ∈ Rd−1 , the equation 

P e iη·y v(xd) = 0 

4 There are several proofs of maximality. The simplest one uses Lax–Milgram theorem.


is a second-order ODE whose solution space has dimension 2n. Ifη = 0, the subspace of solutions 

that vanish at +∞ (stable solutions) has dimension n and is characterized by a first-order 

ODE v ′ = P(0,η)v, where the zero refers to τ = 0 and P(0,η) is the “stable” solution of a 

quadratic matrix equation. To P(0,η), we associate by an explicit formula a Hermitian matrix 

H(η). Then the IBVP is strongly well-posed if, and only if, H(η) is positive definite for 

every η = 0. 

When the boundary data g is non-zero, the situation is not so nice. On the one hand, the 

general IBVP does not fall into a semi-group framework. On the other hand, we usually ask for 

additional boundary terms in estimate (4), both in right-hand side (where g enters) and in lefthand 

side (where the trace of ∇x,tu is estimated). We show here that these strong estimates à la 

Kreiss [10] and Sakamoto [15] never hold in our variational framework, due to the presence of 

surface waves. 

Let us consider for instance isotropic elasticity, where d 3 and the energy is given by 

W(F)= λ 

F + F 

4 

T 2 + μ 

2 (Tr F)2 . (5) 

The parameters λ and μ must satisfy λ>0 and 2λ + μ>0 for strict rank-one convexity. If 

moreover, λ + μ>0, the IBVP admits surface waves (called Rayleigh waves in this case). 

As mentioned above, such waves are incompatible with a strong well-posedness for the nonhomogeneous 

IBVP (1), (2). We shall see that the situation is even worse if λ + μ


2. Convex stored energies 

The role of convex stored energies is fundamental in the study of the homogeneous IBVP. As 

mentioned in the introduction, there is a balance law for the total energy: 

∂t 

 

1 

2 |∂tu| 2 

+ W(∇xu) − 

d 

n 

∂α 

α=1 j=1 

 

∂W 

∂tuj 

∂Fαj 

 

(∇xu) = (∂tu) · f. (6) 

When W is coercive on ˙ 

H 1 (Ω) n , the well-posedness is ensured by the Hille–Yosida theorem and 

Duhamel’s formula, as long as g ≡ 0. Clearly, coerciveness implies strict convexity, although the 

converse is not true. 

The addition of a null-form to W modifies the stored energy W in general, because the integral 

of the minor ∂duj ∂αuk − ∂duk∂αuj over Ω equals a boundary integral. Such an addition, 

although leaving the differential operator P unchanged, does modify the boundary operator B. 

For this reason, when dealing with a specific IBVP, we are only free to add a tangential null-form 

(TNF) to W . This terminology designates the linear combinations of minors Fαj Fβk − FαkFβj 

when 1 α

The assumption tells us that the form 


Q(G) + 2φ(G)· Z +|Z| 2 

takes non-negative values when Z ∈ R l and G has rank one. This amounts to saying that the 

quadratic form 

Q(G) − φ(G) 2 

is non-negative over the cone of rank-one matrices of size (d − 1) × n. Since we have min{d − 

1,n} 2, this implies ([16, Theorem 2.3], see also [18]) that there exists a null-form Q0 over 

M(d−1)×n(R) such that Q+(G) := Q(G) −|φ(G)| 2 + Q0(G) is non-negative. Hence the form 

is non-negative. ✷ 

W(F)+ Q0(G) = Q+(G) + φ(G)+ p(Y) 2 

Example. Let us consider the energy of an isotropic elastic material, given by (5), which is 

convex if and only if λ 0 and 2λ + μd 0, a condition that depends on the dimension. Rankone 

convexity holds if and only if λ 0 and 2λ + μ 0. It is easy to see that in this latter case, 

there exists a null-form Q0 such that W + Q0 is convex; however, this fact is useless when the 

physical domain has a non-trivial boundary, as in our case. The meaningful statement is that 

when λ 0 and λ + μ 0 (an intermediate assumption if d 3)), W(G,Y) is non-negative 

when G has rank one, and therefore there exists a tangential null-form that can be added to W to 

make it convex. For instance, in the extreme case μ =−λ (say that λ = 2), then 

W(F)= 1 

T 

F + F 

2 

2 − 2(Tr F) 2 

= (F12 − F21) 2 + (F23 + F32) 2 + (F13 + F31) 2 + (F33 − F11 − F22) 2 

+ 4(F12F21 − F11F22). (7) 

The last term of the right-hand side is a TNF, and the rest is non-negative. 

2.2. Convex stored energies in general 

For more general energy densities, it may happen that W is convex, even though W is not 

convexifiable in the sense given above. To study the convexity of W in a systematic way, we 

perform a Fourier transform v = Fyu with respect to the tangential variables. This has the effect 

to decouple the tangential frequencies. The following lemma is obvious, except for the notations: 

we extend W to Mn(C) as a sesquilinear form. Thus W keeps real values. Additionally, we use 

the same decomposition F = (G, Y ) as above.


Lemma 2.1. The stored energy W is convex on H 1 (Ω) n if, and only if, the following onedimensional 

quadratic functional Iη is non-negative over H 1 (0, +∞) n , for every vector η ∈ 

R d−1 : 

Notice that when v = Fyu, then 

 

Iη[v]:= 

+∞ 

0 

W(iη⊗ v,v ′ )dxd. 

Fy(∇yu) = iη ⊗ v, Fy(∂du) = v ′ . 

We are thus led to the study of the convexity over H 1 (0, +∞) n of functionals of the form 

I[v]:= 1 

2 

 

+∞ 

0 

w(v,v ′ )dxd, (8) 

where w : C n × C n → R is a sesquilinear form. In addition, the densities w satisfy 

∃ɛ >0 s.t. w(v,iξv) ɛ |η| 2 + ξ 2 |v| 2 , (9) 

because of uniform rank-one convexity. Notice that if W1 and W2 differ only by a TNF, then 

w1 = w2. 

Let us decompose a general density w as 

w(v,v ′ ) =〈Λv ′ ,v ′ 〉+2ℜ〈Av ′ ,v〉+〈Σv,v〉, (10) 

where 〈·,·〉 is the Hermitian product in C n and Λ and Σ are Hermitian positive definite, because 

of (9). We point out that in our variational context, Λ, Σ = Ση and iA = iAη have real entries. 

Here, we give a sufficient condition for the convexity of W, which will be shown necessary 

in Proposition 3.2. 

Theorem 2.2. Let the functional I be defined by (8), (10). Assume that there exists a non-negative 

Hermitian n × n matrix K, with the property that the (2n) × (2n) Hermitian matrix 

 

Σ A+ K 

S := 

A∗ 

(11) 

+ K Λ 

be non-negative. Then I is convex over H 1 (0, +∞) n . 

If S is positive definite, then I is coercive. 

Proof. Let wS be the sesquilinear form associated to S. Then 

I[v]= 1 

2 

+∞ 

 

wS(v, v ′ ) − 2ℜ〈Kv ′ ,v〉 dxd. 

0


Since K is Hermitian, we have 2ℜ〈Kv ′ ,v〉=〈Kv,v〉 ′ , whence 

I[v]= 1 

2 

 

+∞ 

0 

wS(v, v ′ )dxd + 1 

 

Kv(0), v(0) . 

2 

When K and S are non-negative, the right-hand side is non-negative. If, moreover, S is positive 

definite, then I dominates the H 1 -norm. ✷ 

3. Fourier–Laplace analysis 

The most efficient method6 to study the well-posedness of a constant coefficients IBVP in a 

half-space is to perform a Fourier transform in the tangential variables y and a Laplace transform 

in the time variable (see [10,15]). With the notations of the previous section, this yields the ODE 

τ 2 v − Λv ′′ + Aη − A ∗ ′ 

η v + Σηv = 0, (12) 

where we keep track of the dependency of the matrices upon the frequency η. We recall that Λ 

and Σ are symmetric matrices with real entries, and that Aη = iA(η) where A(η) ∈ Mn(R). 

The advantage of the Fourier transform in y is that both the IBVP and the estimate decouple. 

Thus we are led to the equivalent question whether a collection of IBVPs in 1 + 1 dimension is 

strongly well-posed, with a constant C independent of η in the estimate. Each IBVP is obtained 

by replacing ∇y by iη in the differential operator P and the boundary operator B. The reduced 

system is 

∂ 2 t u − Λu′′ + Aη − A ∗ ′ 

η u + Σηu = fη, 

while the reduced boundary condition is 

The needed estimate is 

ˆB(η)u := Λu ′ (0) + A ∗ ηu(0) = 0. (13) 

e −2γT (η ⊗ u, u ′ )(T ) 2 

L 2 + γ 

C 

T 

(η ⊗ u, u ′ )(0) 2 

L 2 + 1 

γ 

0 

e −2γt (η ⊗ u, u ′ )(t) 2 

L 2 dt 

T 

0 

e −2γtfη(t) 2 L2 

dt . (14) 

The estimate above implies classically that the problem (12), together with boundary condition 

Λv ′ (0) + A∗ ηv(0) = 0 and v(+∞) = 0 does not admit a non-trivial solution, when ℜτ >0. 

6 In the sense that it always gives the right result. On specific situations, other methods, from semi-group theory for 

instance, may be faster.


This is the so-called Lopatinskiĭ condition. For such a solution v(xd), one could construct solutions 

uκ(x, t) := e κτt v(κxd) (κ >0), 

whose growth rate κℜτ turns out to be unbounded, revealing a Hadamard instability. 

The estimate (14) actually gives a little bit more, because the boundary frequencies ℜτ = 0 

do play a role in the theory. At least, we easily see that if v satisfies (12) with τ = 0, together 

with Λv ′ (0) + A ∗ η v(0) = 0 and v(+∞) = 0, then u(xd,t):= tv(xd) is a solution of the reduced 

IBVP with fη ≡ 0, u(0) ≡ 0 and u ′ (0) ≡ v. Then taking γ = 1/T and letting T →+∞violates 

(14). We conclude that for the homogeneous IBVP to be strongly well-posed, one needs the 

Lopatinskiĭ property at τ = 0 too. This is a non-trivial fact, related to the special form of the 

problems studied here, since this kind of well-posedness does not need the uniform Lopatinskiĭ 

property. See Section 4.2 for a related look at the point τ = 0. 

Remark that since Λ is positive definite, the energy Iη is coercive for η = 0. By Hille–Yosida, 

the corresponding reduced IBVP is strongly well-posed. Therefore one needs only to consider 

η = 0 in the sequel. 

3.1. The stable subspace 

Assuming that W was strictly rank-one convex, we have at least Λ>0n, thus (12) is genuinely 

of second-order and its solution space has dimension 2n. Actually ∂2 t + P is hyperbolic 

and therefore it is classical that when η and τ vary in such a way that η ∈ Rd−1 and ℜτ >0, then 

the space of solutions of (12) splits into a stable and an unstable subspaces, of which the dimensions 

do not depend on (η, τ). Stable (respectively unstable) means that the elements v of the 

corresponding subspace satisfy v(+∞) = 0 (respectively v(−∞) = 0). When η = 0 and τ = 1, 

the equation reduces to v − Λv ′′ = 0 and therefore both subspaces have dimension n. Inthesequel, 

we denote by E(τ,η) the space of values taken by (v(x), v ′ (x)) when v runs over the space 

of stable solutions. This is an n-dimensional subspace of C2n , called the stable subspace of (12). 

This analysis is valid more generally when (τ, η) runs over a connected set U, provided the 

central subspace remains trivial for every (τ, η) in U. This means that the equation 

det τ 2 In + ξ 2 Λ + iξ Aη − A ∗ 

η + Ση = 0 (15) 

does not have a real root ξ. ThesetUthat we have in mind is the union of the right and left 

half-planes ℜτ = 0, with the set of pairs (iρ, η) defined by ρ ∈ R and |ρ| 0, or (elliptic points of the frequency boundary) τ = iρ 

with ρ ∈ (−h(η), h(η)). 

We now prove that whenever such a τ is the square root of a real number, then E(τ,η) can be 

parametrized by the component v, in the sense that there exists a matrix P = P(τ,η)∈ Mn(C) 

such that v ′ = Pv on E. Since dim E(τ,η) = n, this implies that E(τ,η) is exactly the solution


space of the ODE v ′ = Pv. In particular, P must be a stable matrix in the sense that its spectrum 

has negative real parts. Additionally, P(τ,η)must solve the quadratic equation 

τ 2 In − ΛP 2 + Aη − A ∗ η P + Ση = 0n. (17) 

We begin with 

Lemma 3.1. Let τ be a large enough real number. Then Eq. (17) admits a unique solution P 

among the matrices of which the spectrum has a negative real part (stable matrices). 

This solution has the following properties: 

(1) ΛP + A ∗ is Hermitian. 

(2) P is a (Λ, Σ + τ 2 In)-isometry, in the sense that 

P ∗ ΛP = Σ + τ 2 In. (18) 

Proof. Let ɛ denote 1/τ and let us define Q := ɛP. Then the equation may be rewritten as 

ΛQ 2 = In + 2ɛ(A − A ∗ )Q + ɛ 2 Σ. 

In the limit when ɛ → 0, this equation reduces to Q 2 ∞ = Λ−1 . Since Λ is Hermitian and nonsingular, 

there exists a polynomial T(X) with non-zero simple roots, such that T(Λ −1 ) = 0. 

Any solution of the limit equation satisfies T(Q 2 ∞ ) = 0. The roots of Y ↦→ T(Y2 ) being simple, 

Q∞ is diagonalizable. This tells us that the solutions are of the form Q∞ = U(Λ) where U is 

a polynomial that satisfies U(λ) 2 = 1/λ for every eigenvalue λ of Λ. Since the spectrum of the 

Q∞ must be of non-positive real part, we need U(λ)=−λ −1/2 , that is Q∞ =−Λ −1/2 . 

We now apply the Implicit Function theorem to the non-linear map 

The Q-differential at (0,Q∞) is 

(ɛ, Q) ↦→ ΛQ 2 − In + 2ɛ(A− A ∗ )Q − ɛ 2 Σ, 

C × Mn(C) → Mn(C). 

M ↦→ Λ(Q∞M + MQ∞). 

Since Λ is non-singular and the sum of two eigenvalues of Q∞ may not vanish (it must be 

negative), this differential is non-singular. We thus obtain the existence and uniqueness part. 

We emphasize that the unique stable solution P depends analytically on τ 2 in a neighbourhood 

of infinity. 

Let us define now a matrix R := α + ΛP where α := 1 

2 (A∗ − A) is skew-Hermitian. Then the 

equation can be rewritten as 

(R + α)Λ −1 (R − α) = Σ + τ 2 In. (19) 

When τ is real, the right-hand side is Hermitian. Thus, taking the Hermitian adjoint of (19), we 

obtain 

(R ∗ + α)Λ −1 (R ∗ − α) = Σ + τ 2 In.


This shows that R ∗ is an other solution of (19) and thus P1 := Λ −1 (P ∗ Λ − 2α) is an other 

solution of (17). Since P ∼−τΛ −1/2 ,wehaveP1 ∼−τΛ −1/2 , and therefore the spectrum of 

P1 has a negative real part for τ large enough. From the uniqueness part, we deduce that P1 = P , 

which means R ∗ = R. Hence R is Hermitian. 

Knowing that R is Hermitian, we may recast (19) into (18). ✷ 

Fixing η ∈ R (say η = 0 since the case η = 0 is rather trivial), we consider the maximal 

subinterval Jη = (α, +∞) of (−h(η) 2 , +∞) on which there is an analytical function z ↦→ P(z), 

such that P(z)is a stable solution of (17) with τ 2 = z. Lemma 3.1 ensures that Jη is non-void. By 

analyticity, the properties stated in Lemma 3.1 remain valid over Jη. In particular P(z) remains 

uniformly bounded over this interval, because of (18). 

Differentiating (17) along Jη, wehave 

ΛP dP 

dz 

+ ΛdP 

dz P + (A∗ − A) dP 

dz 

= In. 

Because of Lemma 3.1, part (1), this equation may be written as a Lyapunov equation for the 

unknown ΛP ′ : 

P ∗ Λ dP 

dz 

+ ΛdP 

dz P = In. (20) 

Since P is a stable matrix, this equation has a unique solution, given by 

This formula shows the following monotonicity property. 

Λ dP 

dz =− 

+∞ 

e xP∗ 

e xP dx. (21) 

0 

Lemma 3.2. The map z ↦→ ΛP (z) + A∗ η is monotonous decreasing for the natural order of 

Hermitian matrices. 

Equation (21) may be viewed as an ODE for z ↦→ P(z), with domain the set of stable matrices. 

Notice that every solution of (21) satisfies Eq. (17) for z = τ 2 and some constant matrices A 

and Σ, the latter being Hermitian. To see this, differentiate P(z) ∗ ΛP (z) and deduce that Σ := 

P(z) ∗ ΛP (z) − zIn is constant. Then 

ΛP − (zIn + Σ)P(z) −1 = ΛP (z) − P(z) ∗ Λ 

is skew-Hermitian and constant (differentiate again). Thus there exists an A such that the lefthand 

side equals A − A ∗ . Notice that ΛP (z) + A ∗ is Hermitian. 

From the above remark and Lemma 3.1, it becomes clear that (Jη; z ↦→ P) is the maximal 

solution of the ODE (21) that is defined up to +∞. Because of the bound given by (18), this 

solution remains uniformly bounded and therefore Jη equals (−h(η) 2 , +∞). 

We summarize our results in the following statement.


Theorem 3.1. Assume that W is strictly rank-one convex. When τ 2 ∈ (−h(η) 2 , +∞), the 

quadratic matrix equation (17) admits a unique solution P(τ,η) among stable matrices, with 

the following properties: 

• ΛP + A∗ η is Hermitian, that is ΛP − P ∗Λ = Aη − A∗ η ; 

• P ∗ΛP = Σ + τ 2In; • τ 2 ↦→ ΛP (τ, η) + A∗ η is decreasing for the order of Hermitian matrices. 

Remark. 

• A similar analysis works for the unstable solution Pu of (17). In particular ΛPu + A ∗ is 

Hermitian. However, there is no reason why the other solutions of (17) have this property. 

Likewise, only the stable and the unstable solutions satisfy (18). 

• When τ 2 ∈ (−h(η) 2 , +∞), the stable space is given by the equation v ′ = P(τ,η)v. 

The above analysis also ensures that the limit P0(η) := P(−h(η) 2 ) exists. We discuss in which 

way P0(η) is not a stable matrix. 

Proposition 3.1. The matrix P0(η) admits a pure imaginary eigenvalue iξ0 (ξ0 ∈ R). The associated 

eigenvector X0 can be chosen in R n . Finally, the Hermitian form defined by ΛP0(η) + A ∗ 

vanishes at X0. 

Proof. Since P(z) is a stable matrix for z>−h(η) 2 , the spectrum of P0(η) has a non-positive 

real part. Likewise, the first-order ODE v ′ = P0(η)v defines an n-dimensional invariant subspace 

within the solution space of (12) where τ = ih(η).IfP0(η) was stable, then ih(η) would be an 

interior point of the elliptic interval, which is false. Therefore P0(η) must have a pure imaginary 

eigenvalue, say iξ0, ξ0 ∈ R. LetX0be an associated eigenvector. From Eq. (17), and with Aη = 

iA(η),wehave 

2 T 

ξ0 Λ − ξ0 A(η) + A(η) + Ση − h(η) 2 

In X0 = 0. (22) 

Since the matrix in this equality has real entries, we may choose X0 in Rn . 

Recall that h(η) 2 is the least among the numbers 

2 

λ− ξ Λ + iξ Aη − A ∗ 

η + Ση 

when ξ runs over R. In particular, we have that 

ξ ↦→ g(ξ) := X ∗ 2 

0 ξ Λ + iξ Aη − A ∗ 

η + Ση X0 − h(η) 2 |X0| 2 

is non-negative. Since (22) implies that g(ξ0) = 0, there follows that g ′ (ξ0) vanishes, which gives 

Hence we have 

proving the claim. ✷ 

ξ0X T 0 ΛX0 = X T 0 A(η)X0. (23) 

X ∗ 0 ΛP0(η) + A ∗ X0 = X T 

0 iξ0Λ − iA(η) T X0 = 0,


Remark. The continuity result in Proposition 3.1 is weaker than that obtained by Métivier [14] 

when the characteristics have constant multiplicities. Under the latter assumption, the limit of 

E(τ,η) exists at (ih(η0), η0), without any restriction about (τ, η) ∈ U, while our result concerns 

only the limit along an interval. We recall that points like (ih(η0), η0) are glancing points. 

Proposition 3.1 tells that ΛP0(η) + A∗ η cannot be negative definite. We shall see later on that 

it may be non-positive. In such a case, the identity X∗ 0 (ΛP0(η) + A∗ η )X0 = 0 implies that X0 is 

an eigenvector: 

 

ΛP0(η) + A ∗ η X0 = 0. (24) 

Then Eqs. (24) and (17) yield 

In other words, we have 

3.2. The Lopatinskiĭ determinant 

Ση − h(η) 2 In − ξ0A(η) X0 = 0. (25) 

Λ A ∗ η 

Aη Ση − h(η) 2 In 

iξ0X0 

The boundary operator becomes after Fourier–Laplace transformation 

X0 

ˆB(η)v := Λv ′ (0) + A ∗ η v(0). 

 

= 0. (26) 

For the IBVP to be C∞-well-posed, it is necessary and sufficient that whenever η ∈ Rd−1 and 

ℜτ >0, every stable solution of (12) satisfying ˆB(η)v = 0 vanishes identically (see [9]). This 

is the so-called Lopatinskiĭ condition, which amounts to saying that ˆB(η): E(τ,η) → Cn is an 

isomorphism for all pairs (τ, η) as above. The Lopatinskiĭ condition at point (τ, η) can be written 

as (τ, η) = 0 where the Lopatinskiĭ determinant is defined by 

(τ, η) := det ΛP (τ, η) + A ∗ η , (27) 

provided a stable solution P(τ,η) (necessarily unique) of (17) exists. Because of Theorem 3.1, 

this holds true at least when τ 2 ∈ (−h(η) 2 , +∞). 

Remark. The Euler–Lagrange equations of the extremum problem studied in the next section 

consist of the ODE (12), together with the boundary condition 

Λv ′ (0) + A ∗ ηv(0) = 0, (28) 

that is ˆB(η)v(0) = 0. Therefore the existence of a non-trivial solution v0 given in Theorem 5.1 

implies that ( √ −β,η) = 0. 

From Theorem 3.1, we have the following remarkable property.


Theorem 3.2. With the definition (27), the Lopatinskiĭ determinant is a real analytic function of 

τ 2 over the interval (−h(η) 2 , +∞), continuous over [−h(η) 2 , +∞). 

Remark. The fact that the Lopatinskiĭ determinant is a real-valued function along R + leaves 

the possibility to define a stability index as in [8], even when η = 0. A stability index arises for 

boundary layers or shock layers in viscous systems of conservation laws. This index, as defined 

by Gardner and Zumbrun, concerns the stability of a layer U(xd) under initial disturbances of the 

form δu0(xd); it counts the changes of sign of the so-called Evans function between the origin 

and +∞, using the fact that this function is real-valued when τ ∈ R + and η = 0. Thus it makes 

sense only for one-dimensional systems. It was shown in [8] (one-dimensional case) and in [19] 

(multi-dimensional case) that the Evans function is “tangent” to the Lopatinskiĭ determinant at 

the origin. Thus there is a hope to generalize the stability index to every Fourier frequency, in the 

variational case. 

3.3. Continuation of the Lopatinskiĭ determinant to the hyperbolic region 

We have shown in previous works [4,17] that the generalized stable space E(iρ,η), defined 

as the limit of E(iρ + ɛ,η) as ɛ → 0 + , has a basis made of “real” vectors when ρ is real and 

large enough (hyperbolic region of the frequency boundary). Since the works cited above dealt 

with first-order systems 

∂tu + 

A α ∂αu = 0, 

α 

we need to explain what a “real vectors” means in the context of second-order systems. Since we 

can go back to the first-order by choosing the new variable (∂tu, ∇xu), a “real vector” must be 

understood as a vector of the form 

v ′ 

v 

 

= 

 

iμv 

v 

for some real number μ and real vector v ∈ Rn . 

The property of reality allowed us to define an (equivalent and not canonical) Lopatinskiĭ 

determinant, in such a way that it was a real analytic function of ρ in this region. This property, 

which is true for every hyperbolic IBVP, applies to the class of variational IBVPs. Remark that 

on the contrary, Theorem 3.2 holds true only for the class of variational IBVPs. 

In the latter class, one may wander whether our canonical definition (27) of the Lopatinskiĭ 

determinant is real analytic in the hyperbolic region too, defined by τ = iρ with 

ρ 2 > min 

ξ∈R λ+ 

2 T 

ξ Λ − ξ A(η) + A(η) 

+ Ση . (29) 

We can see that it is true, up to a factor ( √ −1) n . As a matter of fact, in the hyperbolic region the 

matrix P(iρ,η)takes the form iM(ρ,η) where M is a solution of the equation 

ΛM 2 − A(η) + A(η) T M + Σ = ρ 2 In, (30)


with real entries. To see that this is true, we have to show that if ρ satisfies (29), then the 2n roots 

of the polynomial 

X ↦→ det ρ 2 In − X 2 Λ + X A(η) + A(η) T 

− Ση 

are real. Actually, these are the eigenvalues of the matrix 

Conjugation of M0 by 

yields the matrix 

with 

M0 := 

Λ −1 (A(η) + A(η) T ) Λ −1 (ρ 2 In − Ση) 

In 

Λ 1/2 −ξΛ 1/2 

0n 

ξΛ 1/2 

 

, 

 

S ξ−1 T 

M(ξ) := 

ξIn In 

S := Λ −1/2 A(η) + A(η) T Λ −1/2 − ξIn, T := ξS + Λ −1/2 ρ 2 In − Σ Λ −1/2 . 

By assumption, there exists a ξ ∈ R such that T is positive definite. Then conjugation of ξM(ξ) 

by diag{ξIn,T 1/2 } yields the matrix 

ξS ξT 1/2 

ξT 1/2 ξIn 

The latter, being Hermitian, is diagonalizable with a real spectrum. This proves the claim, and 

the fact that every solution of (30) belongs to Mn(R). Of course, only that one associated to the 

eigenvalues iμ which enter into the right half-plane when ℜτ increases is relevant. 

Let us denote this matrix by M(ρ,η). Then definition (27) can be extended, and yields 

 

. 

0n 

(iρ, η) = i n det ΛM(ρ, η) − A(η) T , 

where the matrix in the determinant has real entries, hence the determinant is a real number. We 

leave open the fact that the matrix P(τ,η) can be defined continuously and analytically along a 

neighbourhood of the real axis within the right half-plane. In this case, we should have a unique 

analytic function, real-valued along R and along (−ih(η), ih(η)), and real-valued up to the factor 

i n in the far field of the imaginary axis. A typical example of such a function is τ ↦→ √ τ 2 + 1, 

which is real for τ ∈ (−i, +i) and pure imaginary in the rest of the imaginary axis. 

 

.

3.4. The zeroes of ( √ z, η) 


Theorem 3.3. The function z ↦→ ( √ z, η) vanishes at least once in [−h(η) 2 , +∞), and at most 

n − 1 times in (−h(η) 2 , +∞). The largest zero is the number a such that ΛP (a) + A ∗ is nonpositive 

and singular. 

If ΛP (0) + A ∗ has a positive eigenvalue, then the IBVP is strongly unstable. 

Remark. In particular, the so-called uniform Lopatinskiĭ condition is never satisfied, except in 

one space dimension, since (τ, η) must vanish somewhere in ℜτ 0, (τ, η) = (0, 0). This 

implies that the estimates in the non-homogeneous IBVP (g ≡ 0 in (2)) do suffer a loss of derivatives. 

Proof. The signature of ΛP (z) + A∗ varies monotonically and ( √ z, η) vanishes only when 

this signature changes. Thus it may vanish at most n times in (−h(η) 2 , +∞), and this maximum 

is achieved when ΛP0(η) + A∗ η is positive definite. However, Proposition 3.1 rules out this 

possibility. Likewise Proposition 3.1 prevents ΛP0(η) + A∗ η to be negative definite, whence the 

existence of one zero at least in [−h(η) 2 , +∞). 

By continuity and monotonicity, ( √ z, η) does not vanish in (a, +∞) when ΛP (a) + A∗ is non-positive. The last claim is the special case a = 0; under the assumption, the Lopatinskiĭ 

condition is not satisfied. ✷ 

3.5. The converse to Theorems 2.2 and 3.3 

Let S(K) be the matrix occurring in Theorem 2.2. We begin with the observation that if we 

choose K =−(ΛP (a) + A ∗ ), which is Hermitian, then the identities (17) and (18) imply that 

 

P(a) ∗ 

S(K) = Λ 

−In 

 

P(a),−In − a 

which is positive definite when a0n 

and S>02n in the computation above. ✷ 

This proposition has a remarkable consequence for the characterization of strongly well-posed 

homogeneous IBVP. We recall that a necessary (though not sufficient) condition for the strong 

well-posedness is that the Lopatinskiĭ property be satisfied for every pair (τ, η) with either 

ℜτ >0orτ = 0. In particular, it must hold true when τ ∈ R + . We want to prove a converse 

 

,


of the last sentence. For this, let us assume only that (·,η) does not vanish over R + . Applying 

Proposition 3.2 and Theorem 2.2, we see that the functional I defined by (8) is convex and coercive 

over H 1 (R + ). Thus we may apply the Hille–Yosida theorem and obtain the well-posedness 

of the homogeneous IBVP at fixed η, in1+ 1 dimension. In turn, this well-posedness ensures 

that the Lopatinskiĭ property holds true for every ℜτ >0. We summarize this analysis in the 

following statement. 

Theorem 3.4. Assume that (·,η)does not vanish over R + . Then 

Iη[v]:= 1 

2 

 

+∞ 

is convex and coercive over H 1 (R + ). In particular: 

0 

w(v,v ′ )dxd 

• The corresponding homogeneous IBVP at frequency η is well-posed in H 1 (R + ). 

• ˆB(η): E(τ,η) ↦→ C n is an isomorphism when ℜτ >0. 

• The same holds true in the non-elliptic part of the boundary ℜτ = 0. 

To our knowledge, it is the first time that a structural assumption yields the conclusion that a 

strong instability (the vanishing of (τ, η) for some ℜτ >0, or more generally the Lopatinskiĭ 

condition) must happen either for a real τ , or nowhere. We point out that Theorem 3.4 is a kind 

of converse to Theorem 3.3, in the sense that if the homogeneous IBVP is strongly unstable, 

then (·,η) must have a zero over R + , and therefore ΛP (0) + A ∗ must have a non-negative 

eigenvalue (presumably positive). 

3.5.1. Well-posedness of the full homogeneous IBVP 

We now let varying the space frequency η. Looking for a criterion of well-posedness for the 

full homogeneous IBVP (1), (2) (thus with g ≡ 0), we make the natural assumption that does 

not vanish at all over R + × R. As shown above, each Iη is convex and coercive over H 1 (R + ). 

Let us remark that the various objects of the theory are homogeneous in their arguments. If κ is 

a positive real number, then we have 

A(κη) = κA(η), Σκη = κ 2 Ση, E(κτ,κη)= E(τ,η), 

P(κτ,κη)= κP (τ, η), h(κη) = κh(η), (κτ, κη) = κ(τ, η). 

Using these properties and the fact that the unit sphere of R d−1 is compact, we see that the 

number a = a(η) in the construction above can be chosen in the form a(η) =−ɛ|η| 2 ,forafixed 

ɛ>0, small enough. We deduce that 

+∞ 

′ 2 2 2 

Iη[v] Cɛ |v | +|η| |v| dxd, (32) 

0 

where C does not depend on η. Then inverse Fourier transform gives that W dominates the norm 

of H˙ 1 (Ω). In conclusion, W is convex and coercive over H˙ 1 (Ω). Whence our main result:


Theorem 3.5. We assume that W is strictly rank-one convex. Then the following statements are 

equivalent to each other: 

(1) The stored energy W is convex and coercive over H˙ 1 (Ω). 

(2) The homogeneous IBVP is strongly well-posed in H˙ 1 (Ω). 

(3) One has (τ, η) = 0 for every τ ∈ R + , η ∈ R \{0}. 

(4) For every η = 0, the Hermitian matrix H(η):= −(ΛP (0,η)+ A∗ η ) is positive definite. 

Proof. Essentially everything has been proved yet. We content ourselves to recall the main arguments. 

(1) ⇒ (2). Apply Hille–Yosida theorem. 

(2) ⇒ (3). The strong well-posedness implies the Lopatinskiĭ condition for ℜτ >0, whence 

the special case of τ ∈ (0, +∞). We have seen the necessity of Lopatinskiĭ forτ = 0inthe 

introduction of Section 3. 

(3) ⇒ (4). The assumption tells that the Hermitian matrix ΛP (τ, η) + A∗ η is non-singular for 

every τ ∈ R + . Since it is negative definite for τ large, it remains so for every τ 0, by continuity. 

(4) ⇒ (1). In the present paragraph, we have shown that the assumption implies a uniform 

inequality (32) for some ɛ>0, at least for η = 0. By continuity, it remains true for η = 0, whence 

the coercivity of W. ✷ 

Remark. The matrix K(η) constructed above is homogeneous of degree one in η, but is not 

linear in η in general. There is no special reason why a single non-tangential null-form could be 

added to W in such a way that the new density and the boundary term be strictly convex. 

3.6. The influence of non-tangential null-forms to W 

We observe on the one hand that the matrix P(z)depends only on z, Ση and ℑAη = i 2 (A∗η − 

Aη) = 1 2 (A(η) + A(η)T ), but not on ℜAη, while the matrix ΛP (τ 2 ) + A∗ η that enters in the 

Lopatinskiĭ determinant does involve ℜAη. On the other hand, the addition of a non-tangential 

null-form is a mean for modifying ℜAη and only that. This remark is consistent with that made 

above: the addition of a non-tangential null-form does not change the differential operator, but 

modifies the boundary operator. 

Recall that the matrix A(η) has real entries. Thus 

ℜAη = i T 

A(η) − A(η) 

2 

 

involves only the skew-symmetric part of A(η). Of course, h(η) depends on the symmetric part 

of A(η), but not on its skew-symmetric part. It turns out that the non-tangential null-forms are 

in one-to-one correspondence with the skew-symmetric matrices. Therefore playing with nontangential 

null-forms allows us to add to ΛP + A ∗ an arbitrary matrix of the form iM, with 

M ∈ Skewn(R). 

Proposition 3.3. Given Λ, Ση and the symmetric part of A(η), one can choose the skewsymmetric 

part in such a way that (±ih(η), η) vanishes. 

In other words, given the energy density, one can add a null-form in such a way that 

(±ih(η), η) vanishes.


Proof. Let Y denote the real vector (ξ0Λ − A(η) T )X0. From (23), Y is orthogonal to X0. Thus 

there exists a skew-symmetric matrix M with real entries such that MX0 = Y . If we replace A(η) 

by Ã := A(η) − M, we obtain (ΛP0(η) + Ã ∗ )X0 = 0, whence 

as desired. ✷ 

Remark. 

det ΛP0(η) + Ã ∗ = 0, 

• There is no reason why the additional null-form of the proposition be independent of η. Thus 

it is not possible in general to find a single null-form such that (±ih(η), η) = 0 for every η. 

This could be achieved by adding a null-form, pseudo-differential in the tangential variables. 

• One may wander whether it is possible to adjust the skew-symmetric part of A(η) in such a 

way that ΛP0(η) + A∗ η be non-positive. In this case, (√z, η) vanishes only at the extremity 

−h(η) 2 , whence a well-posedness at frequency η, plus the property that the surface waves 

have an infinite energy! We leave the reader check that such a choice is possible if, and 

only if, the restriction of the Hermitian form X ↦→ X∗ (ΛP0(η) + A∗ η )X to real vectors is 

non-positive. 

4. Surface waves 

Assume that a given hyperbolic IBVP satisfies the Lopatinskiĭ condition, though not necessarily 

in a uniform way. Let us assume also that the stable space E(τ,η) admits a continuous 

extension E(iρ,η) at some boundary point (iρ, η). We say that the IBVP admits a surface wave 

with frequency (ρ, η) if ˆB(η): E(iρ,η) → C n is singular. This amounts to the existence of a 

non-trivial solution v of (12) for τ = iρ, with the properties that ˆB(η)v(0) = 0 and that v is 

polynomially bounded at infinity. Then 

u(x, t) := e i(ρt+η·y) v(xd) 

is a solution of (1), (2) with f ≡ 0 and g ≡ 0. When v tends to zero at +∞, the decay is exponential 

and we say that the surface wave has finite energy. When E(iρ,η) equals the stable subspace 

of (12), a surface wave at frequency (ρ, η) is automatically of finite energy. This is, of course, 

the case when (iρ, η) lies in the elliptic region of the frequency boundary. Thus the existence of 

a surface wave of elliptic frequency (ρ, η) is equivalent to (iρ, η) = 0. Theorem 3.3 tells that 

if W is convex and coercive, a finite energy surface wave (FESW) must exist at space frequency 

η for some time frequency ρ, except in the marginal case where ΛP0(η) + A∗ η is non-positive. 

A well-known example of FESW is the Rayleigh wave in isotropic elasticity. 

See [3] for an analysis of the role of surface waves in the homogeneous IBVP. It is also 

explained in [4] that among the class of strictly (or symmetric) hyperbolic IBVPs, the problems 

that admit FESW are non-generic. They form a hypersurface in the space of hyperbolic IBVPs: 

under a small perturbation in the differential operator, or in the boundary operator, the IBVP falls 

either in the strongly well-posed class or in the strongly ill-posed class, generically. We shall 

show below that this picture becomes false when the perturbation keeps our IBVP within the 

class of variational problems (1), (2). Of course, this restriction is non-generic in the context of 

[4], but does have a physical relevance.

4.1. Persistence of surface waves 


Let us assume that for a given strictly rank-one convex energy W0, the IBVP (1), (2) admits a 

surface wave of finite energy for some frequency η. Then there exists a ρ0 ∈ (−h(η), h(η)) such 

that (iρ0,η)= 0. Because of Lemma 3.2, there is a change in the signature of ΛP (iρ, η) + A∗ η 

across ρ0. Say that for ɛ>0 small enough, ΛP (i(ρ0 ± ɛ),η) + A∗ η are non-singular, and the 

numbers p± of their positive eigenvalues satisfy p+ >p−. 

When we make a small enough perturbation W = W0 + δW, the Hermitian matrices 

ΛP (i(ρ0 ± ɛ),η) + A∗ η still have p± positive eigenvalues. Therefore there exist a ρ in (ρ0 − ɛ, 

ρ0 + ɛ) such that ΛP (iρ, η) + A∗ η be singular. Then (iρ, η)) = 0 and there exists a FESW at 

frequency (ρ, η). This is the phenomenon of persistence of FESW. Remark that there may be 

(and there are, generically) p+ − p− such roots ρ in the interval (ρ0 − ɛ,ρ0 + ɛ). Notice that the 

same kind of persistence occurs when we perturb the frequency η (considering instead η + δη) 

instead of the energy density. 

Let us write η0 instead of η, and let us vary the space frequency. When p+ = p− + 1, ρ0 is 

a simple root of (·,η0) in the unperturbed problem. Then there exists an analytic function 

η ↦→ ρ(η) defined in a neighbourhood of η0, satisfying ρ(η0) = ρ0 and (ρ(η), η) = 0. If ρ 

is globally defined, then the wave front of the trace of solutions of (∂ 2 t + P)u∈ C∞ , along the 

boundary {xd = 0}, is included in the characteristic cone 

(ρ, η) ∈ R × R d−1 ; ρ = ρ(η) . 

Finally, we point out that the dimension of the space of FESWs is related to (p−,p+): 

Proposition 4.1. With the notations above, the dimension of the kernel of ΛP (iρ0,η)+ A∗ η , that 

is the dimension of the space of FESWs at frequency (ρ0,η), equals p+ − p−. 

Proof. When ρ → ρ0 − 0, ΛP (iρ, η) + A ∗ η has p− positive and n − p− negative eigenvalues. 

Since this Hermitian matrix increases when ρ increases (notice that ρ ↦→ z = (iρ) 2 is decreasing!), 

we deduce by continuity that ΛP (iρ0,η)+A ∗ η has exactly p− positive eigenvalues. Letting 

ρ → ρ0 + 0, we find likewise that ΛP (iρ0,η) + A ∗ η has exactly n − p+ negative eigenvalues. 

The dimension of its kernel being equal to the multiplicity of the null eigenvalue, it is 

n − p− − (n − p+) = p+ − p−. ✷ 

4.2. Transition to instability 

From Theorem 3.5, a transition towards instability happens precisely when the matrix H(η)= 

−(ΛP (0,η)+ A∗ η ) stops to be positive definite. This can be achieved for instance by choosing 

ℜAη so as to have X∗ (ΛP (0) + A∗ η )X > 0 for some vector X, following an idea employed in 

Section 3.6. 

Of course, since the IBVP is ill-posed whenever there exists at least one frequency η for 

which (·,η)has a root of positive real part, this transition happens when the sign of 

min d−1 

λ− H(η) : η ∈ S 

changes.


We now enlight a connection between the transition described above and the nature of the 

elliptic BVP 

Lu = f in Ω, (33) 

Bu = g on ∂Ω. (34) 

Such problems have been studied systematically by Agmon et al. [1]. See also the seminal work 

by Lopatinskiĭ [11]. They show that a priori estimates are available if and only if a complementing 

condition holds true at non-zero frequencies η. It turns out that this condition coincides with 

(0,η)= 0 for all non-zero vector η ∈ R d−1 , where is the Lopatinskiĭ determinant. Therefore, 

if a transition between well- and ill-posedness occurs at some hyperbolic IBVP, then the 

corresponding elliptic BVP is ill-posed. 

We warn the reader that the converse is not true when d 3, since the set of ill-posed elliptic 

BVPs has a non-void interior (in the set of parameters). As a matter of fact, in most cases, 

the vanishing of the function η ↦→ (0,η) at some point η0 = 0 means that the sign of this 

function changes. 7 Then a small perturbation in L or B yields a small perturbation in , so that 

(0, ·) still vanishes somewhere and the modified elliptic BVP remains ill-posed. Within the 

set of ill-posed elliptic BVPs in the sense of ADN, only those at the boundary may correspond 

to a transition in the hyperbolic IBVP. More precisely, the complement function must vanish 

somewhere, without changing sign. 

We point out that this analysis leaves the possibility that the hyperbolic IBVP is ill-posed 

while the steady elliptic BVP is well-posed. This possibility is confirmed by explicit examples; 

see, for instance, Theorem 6.1. 

5. An extremum problem 

We solve in this section an abstract problem of extremum. Theorem 5.1 gives an alternate 

proof of the fact that every variational IBVP either admits surface waves, or is unstable in the 

Hadamard sense. 

We start with a functional I as in (8), where w is a sesquilinear form satisfying (9). Let us 

define the finite number 

I[v] 

β := inf , E[v]:=1 

v≡0 E[v] 2 

 

+∞ 

0 

|v| 2 dxd. (35) 

We have the property that I − γE is convex over H 1 (R + ) if and only if γ β. Testing with the 

fields v of the form e −ωxd V , we immediately find the property 

In particular, we have 

(ℜω 0) ⇒ βIn Θ(ω) , Θ(ω):= |ω| 2 In − ωA −¯ωA ∗ + Σ. (36) 

βIn Θ(iξ), ∀ξ ∈ R. 

7 This is, of course, not the case if d = 2, because of homogeneity.


When the latter inequalities are strict, a fact that occurs for instance when 

min 

ℜω0 λ− 

 

Θ(ω) < min 

ξ∈R λ− 

 

Θ(iξ) , (37) 

we have the following result. 

Theorem 5.1. Let us assume (9), together with the strict inequality 

β


for every z ∈ H 1 (R), a property that follows easily from Fourier analysis. We thus have I[zm] 

λ0E[zm]. Passing to the limit in (39), we deduce 

β 

λ0ℓ 

lim I[Vm]+ 

2 2 . 

Since I + rE is convex, it is weakly lsc over H 1 (R + ). Using the fact that Vm converges H 1 - 

weakly and L 2 -strongly towards v0, we conclude that 

β − λ0ℓ 

I[v0] lim I[Vm] . 

2 

Finally, with I[v0] βE[v0]=β(1 − ℓ)/2, we obtain 

ℓ(λ0 − β) 0. 

The assumption (38) then implies ℓ = 0, meaning that vm converges towards v0 strongly in 

L 2 (R + ). ✷ 

To complete the proof, we have to prove Lemma 5.1. For that purpose, we may assume that 

vm converges towards v0 in the following senses: weakly in H 1 (R + ), strongly in L 2 loc (R+ ) and 

almost everywhere. Let ψ ∈ C ∞ (R) be such that ψ(y) ≡ 0fory −1 and ψ(y)≡ 1fory 1, 

and 0 ψ 1 everywhere. Define also χ := 1 − √ 1 − ψ. We shall choose 

Vm(x) := 1 − ψ(x − xm) 1/2 vm(x), zm(x) := χ(x − xm)vm(x), 

for a suitable sequence xm. We notice first that Vm → v0 almost everywhere provided 

lim xm =+∞. (40) 

This implies that Vm converges weakly in L 2 towards v0. In order to have the strong convergence, 

we only need that E[Vm]→E[v0], which is equivalent to 

 

lim 

+∞ 

0 

Finally, the difference I[vm]−I[Vm]−I[zm] can be recast into 

ψ(x − xm) vm(x) 2 dx = 1 − 2ℓ. (41) 

 

xm+1 

xm−1 

B(Vm,zm)dxd, 

where B is a bilinear form with smooth coefficients, which involves the arguments and their first 

derivatives. In order to satisfy (39), it is enough to have 

lim 

 

xm+1 

xm−1 

 

v ′ 

 

m 

2 +|vm| 2 dxd = 0. (42)


There remains to choose xm so as to satisfy all of (40)–(42). To do so, we employ the diagonal 

extraction to build a subsequence, still denoted by (vm)m1, such that 

From (43) and 

we infer that 

and therefore 

m 

0 

m 

lim |vm − v0| 2 dxd = 0. (43) 

0 

|vm| 2 m 

 

dxd − 2ℓ =ℜ 〈vm − v0,vm + v0〉 dxd − 

A trivial consequence of (43) is that 

0 

0 

+∞ 

m 

|v0| 2 dxd, 

m 

lim |vm| 2 dxd = 2ℓ, (44) 

 

lim 

+∞ 

m 

0 

|vm| 2 dxd = 1 − 2ℓ. 

am 

lim |vm − v0| 2 dxd = 0 

for every sequence such that am


On the one hand, condition (41) is certainly satisfied for xm ∈ (am + 1,m− 1). On the other 

hand, the length of this interval tends to +∞, while the integral 

 

m−1 

am+1 

 

v ′ 

m 

2 +|vm| 2 dxd 

remains bounded (as it is when we integrate over R + ). Thus there exists a subinterval of length 

two on which the integral is less than M/(m − am), thus tends to zero. We can choose xm the 

center of this subinterval to satisfy condition (42). 

Note. The calculations of Section 3.6 show that assumption (38) is often satisfied. 

6. Examples 

We point out that in most examples, we are able to compute E(τ,η) even for non-real τ 2 , 

thanks to the rather simple structure of the energy density. Remark that the Lopatinskiĭ determinants 

computed below are not those given by formula (27), since the explicit computation of the 

matrix P may be cumbersome. We have followed the more classical way, where we compute an 

explicit basis of the stable subspace. This method has the disadvantage that such a set of “incoming 

modes” may fail to be a basis at some frequency pairs (τ, η), whence spurious zeroes 

of . 

6.1. 2-dimensional wave-like systems 

Let us fix n = d = 2. The density 

W0(F ) := 1 

|F |2 

2 

yields the differential operator P = x ⊗ I2. Equation (12) is 

u ′′ = τ 2 + η 2 u, 

and its stable subspace is given by 

u ′ =−ω(τ,η)u, ω(τ,η) := 

 

τ 2 + η2 , 

where the square root is that of positive real part. We recognize P(τ,η)=−ω(τ,η)I2, whence 

P(0,η)=−|η|I2. In particular, h(η) =|η|. 

Since A ≡ 02 and Λ = I2, wehave 

ΛP (τ, η) + A ∗ η =−ω(τ,η)I2. 

This is a negative real number when τ 2 ∈ (−h(η) 2 , +∞). Hence ( √ z, η) vanishes only at the 

extremity z =−h(η) 2 . In conclusion, the IBVP is well-posed, with surface waves of infinite 

energy.


If we add to W0 a null-form, we do not change the PDEs but we change the boundary operator 

B. We thus consider the density 

Wκ(F ) := W0(F ) + κ det F. 

The boundary operator associated to Wκ is thus 

 

∂2u1 − κ∂1u2, 

u ↦→ 

. 

∂2u2 + κ∂1u1 

Using Fourier–Laplace variables, we deduce the Lopatinskiĭ determinant 

 

−ω −iκη 

(τ, η; κ) = det 

= τ 

iκη −ω 

2 + 1 − κ 2 η 2 . 

As expected, (·,η; κ) is real analytic, with D(z,η; κ) = z + (1 − κ 2 )η 2 . When η = 0, the root 

(κ 2 − 1)η 2 of D(·,η; κ) hasthesamesignasκ 2 − 1. Therefore the hyperbolic IBVP is ill-posed 

for |κ| > 1, despite the fact that the corresponding elliptic BVP is well-posed within this range 

of parameter (the effect of d = 2). 

The transition towards instability occurs when κ =±1. Say that κ = 1. The corresponding 

elliptic BVP consists in xu = f for x2 > 0, together with the boundary conditions ∂2u1 − 

∂1u2 = g1 and ∂2u2 +∂1u1 = g2. This is the Cauchy problem for the Cauchy–Riemann equations 

in terms of the complex function γ := ∂1u2 − ∂2u1 + i(∂2u2 + ∂1u1): 

∂γ 

= f, γ = g. 

∂ ¯z 

This is the most known ill-posed problem. 

6.2. 3-dimensional wave-like systems 

In order to have a non-trivial family of ill-posed elliptic BVPs, we take n = d = 3 and we 

form an energy density through the same strategy as above: 

W(∇xu) := 1 

2 |∇xu| 2 + ∂3u1V ·∇yu2 − ∂3u2V ·∇yu1 + ∂3u2Y ·∇yu3 − ∂3u3Y ·∇yu2 

+ ∂3u3Z ·∇yu1 − ∂3u1Z ·∇yu3. (45) 

Here, vectors V,Y,Z∈ R2 are given and play the role of parameters. The differential operator is 

L =−x ⊗ I3. The incoming modes are given by the equation 

v ′ 

=−ωv, ω = τ 2 +|η| 2 , 

whence P(τ,η)=−ω(τ,η)I3. The boundary operator is 

⎛ 

⎞ 

∂3u1 + V ·∇yu2 − Z ·∇yu3 

u ↦→ ⎝ ∂3u2 + Y ·∇yu3 − V ·∇yu1 ⎠ , 

∂3u3 + Z ·∇yu1 − Y ·∇yu2


from which we find 

v ∗ H(η)v=|η||v| 2 + 2Y · ηℑ(v3 ¯v2) + 2Z · ηℑ(v1 ¯v3) + 2V · ηℑ(v2 ¯v1). (46) 

The Lopatinskiĭ determinant writes 

with 

 

 

−ω iV · η −iZ · η 

 

 

(τ, η) = 

−iV · η −ω iY · η 

 

 

 

iZ · η −iY · η −ω 

= ωq(η)− ω 2 , 

q(η) := (V · η) 2 + (Y · η) 2 + (Z · η) 2 . (47) 

Let λ±(q) denote the eigenvalues of the quadratic form q. The roots of (·,η) are given by 

τ =±i|η| (for ω = 0), and by 

τ 2 = q(η)−|η| 2 . (48) 

On the one hand, this equation has a positive real root for some η if and only if 1 1. 

When λ+(q) < 1, the well-posedness of the hyperbolic IBVP is a consequence of Theorem 

3.5. Alternately, we may remark that W is convexifiable in the sense of Section 2.1, and 

then apply the Hille–Yosida theorem. Thanks to Theorem 2.1, and since d = 3,weonlyhaveto 

verify that W(G,F3·) is positive whenever G ∈ M2×3(R) has rank one. To check this property, 

we rewrite 

where B is the boundary operator and 

2W(∇xu) =|Bu| 2 + W1(∇yu), 

W1(∇yu) =|∇yu| 2 − (V ·∇yu2 − Z ·∇yu3) 2 − (Y ·∇yu3 − V ·∇yu1) 2 

− (Z ·∇yu1 − Y ·∇yu2) 2 .

When G = η ⊗ r, we obtain 

Because of |r × σ | |r||σ |, we deduce 


W1(η ⊗ r) =|η| 2 |r| 2 2 Y · η 

 

− r 

× Z · η . 

 

V · η 

W1(η ⊗ r) |η| 2 − q(η) |r| 2 , 

which is positive if λ+(q) 1. Whence W is convexifiable. 

6.3. Isotropic elasticity 

We turn now to a practical application in elasticity, where again n = d = 3. We consider the 

case of an isotropic stored energy density given by (5). We recall that W is convex if λ 0 and 

2λ + 3μ 0, and that it is convexifiable by a TNF if λ 0 and λ + μ 0, the latter condition 

being weaker than the former. 

The differential operator. The PDEs for x3 > 0are 

∂ 2 t 

u = λu + (λ + μ)∇ div u. (49) 

Let us perform the Laplace–Fourier transform. In the new unknown, we distinguish the tangential 

components v⊥ := (v1,v2) and the normal component vd. Because of isotropy, it is 

enough to work with the scalar quantity w := iv⊥ · η: 

τ 2 w = λw ′′ − (2λ + μ)|η| 2 w − (λ + μ)|η| 2 vd, 

τ 2 vd = (2λ + μ)v ′′ 

d − λ|η|2vd + (λ + μ)w ′ . 

Let us assume that ℜτ >0, so that τ 2 ∈ C \ R − . When looking for modes in exp(−ωxd), we 

find that ω ∈{ωP ,ωS}, where 

and 

ωP,S := 

τ 2 

c2 +|η| 

P,S 

2 

cP := 2λ + μ, cS := √ λ 

are the velocities of the pressure ans shear waves. These are the waves propagated by Eq. (49). 

They are distinct provided λ + μ = 0 and τ = 0. Notice that cP is larger than cS in the convexifiable 

case, and smaller otherwise. When either λ + μ = 0orτ = 0, we have ωP = ωS, and there 

is an additional mode of the form (axd + b)exp(−ωxd).


Pressure waves and shear waves. Simple calculations show that the mode associated to ωP is 

proportional to the vector 

 

w,vd,w ′ ,v ′ 

d P := |η| 2 ,ωP , −ωP |η| 2 , −ω 2 

P , 

and that the mode associated to ωS is proportional to the vector 

 

w,vd,w ′ ,v ′ 

d S := ωS, 1, −ω 2 

S , −ωS . 

The boundary condition. In this variational context, the boundary condition is that of zero 

normal stress: 

λ(∂du +∇ud) + μ(div u)ed = 0, x3 = 0. 

After the Fourier–Laplace transformation, we find the equivalent system 

w ′ −|η| 2 vd = 0, 

(2λ + μ)u ′ d 

+ μw = 0. 

The Lopatinskiĭ determinant. We now write that (w, vd,w ′ ,v ′ d ) is a linear combination of the 

pressure and shear modes: 

 

w,vd,w ′ ,v ′ 

d = α w,vd,w ′ ,v ′ 

d S + βw,vd,w ′ ,v ′ 

d P . 

Writing the boundary condition, we obtain a 2 × 2 linear system, of which the determinant is the 

Lopatinskiĭ determinant: 

 

ω 

(τ, η) = 

 

2 S +|η|2 2|η| 2ωP −2λωS μ|η| 2 − (2λ + μ)ω2 

 

 

 

P 

. 

This gives 

λ −1 (τ, η) = 4|η| 2 

ωSωP − 2|η| 2 + 

This formula extends by continuity when ℜτ = 0. 

Discussion. If vanishes, then 

16|η| 4 

 

|η| 2 + 

τ 2 

c 2 S 

 

|η| 2 + 

an equation that can be recast into Q(τ 2 /|η| 2 ) = 0, where 

τ 2 

c 2 P 

 

= 2|η| 2 + 

 

z 

Q(z) := 

c2 4 

z 

+ 2 − 16 

S 

c2 

z 

+ 1 

S c2 P 

τ 2 2 . (50) 

λ 

τ 2 

c 2 S 

 

+ 1 . 

4 

,

This polynomial satisfies 


Q −c 2 

′ 1 

S = 1, Q(0) = 0, Q (0) = 16 

c2 − 

S 

1 

c2 

, Q(±∞) =+∞. 

P 

If λ + μ is positive, W is convexifiable by a TNF, and therefore the homogeneous hyperbolic 

IBVP is well-posed. This can be checked directly by showing that does not vanish, except at 

boundary points of elliptic type τ =±icR|η|, where −c2 R is the unique root of Q in the interval 

(−c2 S , 0). Notice that Q′ (0) is positive in this case, so that such a root must exist. The number 

cR, with the dimension of a velocity, is the speed of Rayleigh waves, the FESW of this problem. 

If λ + μ is negative, then Q ′ (0) is negative, and Q must have a positive root c2 ,bythe 

Intermediate Value theorem. For τ = c|η|, we then have 

 

4|η| 2 

ωSωP + 2|η| 2 + 

τ 2 2 (τ, η) = 0, 

λ 

where the parenthesis is a positive real number. Therefore (c|η|,η) ≡ 0 and the hyperbolic 

IBVP is strongly ill-posed. 

Theorem 6.2. When n = d = 3 and the energy density is given by (5), with λ>0 and 2λ + μ>0 

for uniform rank-one convexity, we have: 

(1) The homogeneous hyperbolic IBVP is well-posed provided λ + μ>0, or equivalently 

cP >cS. 

(2) The hyperbolic IBVP is strongly ill-posed when λ + μ0, certainly give a control of the L2-norms of ∂iui (i = 1,...,3), ∂1u2 and ∂2u1, and 

of ∂2u3 + ∂3u2, ∂1u3 + ∂3u1. But since Ω has a boundary, Korn’s inequality does not apply 

and this does not give a control neither of ∂3u nor of ∇u3. As remarked in Section 3.5.1, the 

coercivity of W comes from a corrector that is neither a null-from (it does yield a boundary 

integral), nor differential (the matrix K does depend on η). Following Section 3.5, we may 

take 

 

λ|η|I2 i(ν − λ)η 

K(η) = 

i(λ− ν)ηT 

λ|η|


with ν := c0 min{λ,λ+ μ} and c0 is some suitable positive number, independent of the other 

parameters. 

The steady elliptic BVP. Because of the coincidence between ωP and ωS when τ = 0, the 

calculation above does not give an answer for the steady BVP. This is due to our choice of the 

modes, because we do not obtain the affine modes in the limit; it has the effect that becomes 

trivial in this limit case. A more careful analysis would give us a non-trivial function . 

Here, the differential equations imply 

2 

∂d −|η| 22 w = 0, 

and the same for vd. The modes that decay at +∞ thus satisfy 

 

∂d +|η| 2 

w = 0, ∂d +|η| 2 vd = 0. 

In other words, we have w = (αxd + β)exp(−|η|xd), with an analogous formula for vd. Going 

back to the very ODEs, we find the general formula for the decaying modes: 

w = (λ + μ)axd + b |η| 2 e −|η|xd , 

vd = (λ + μ)a|η|xd + (3λ + μ)a + b|η| e −|η|xd . 

Inserting this into the boundary conditions, we obtain again a linear 2 × 2 system, whence a 

correct Lopatinskiĭ determinant 

(0,η)=−4λ(λ + μ)|η| 4 . 

This shows that the elliptic BVP is well-posed if and only if λ + μ = 0. 

Remark that the ill-posed elliptic BVPs form a hypersurface in the space of parameters (λ, μ), 

despite the dimension three of the physical domain. This is due to the isotropy assumption. For 

a non-isotropic energy density, we expect that these ill-posed steady problems form a set of 

non-void interior. Then the transition between well-posed and ill-posed hyperbolic IBVPs would 

occur along a boundary of this set. 

6.4. Phase transition in a van der Waals fluid 

Kreiss’ and Sakamoto’s theory of hyperbolic IBVPs has been adapted by A. Majda [12] to 

treat the linearized stability of shock waves in systems of conservation laws (a second paper 

[13] treats the non-linear stability). The context differs slightly, in the sense that the boundary 

condition is coupled with a PDE that describes the evolution of the shock front. These boundary 

conditions come from the linearization of the Rankine–Hugoniot condition. Despite these 

technical differences, the same notions of Kreiss–Lopatinskiĭ condition, UKL and Lopatinskiĭ 

determinant remain relevant. 

Majda’s method is relevant for Lax shocks, where the Rankine–Hugoniot condition provides 

the right number of boundary conditions. It has been adapted by H. Freistühler [7] to 

so-called undercompressive shocks, when additional jump conditions are given in complement 

of the Rankine–Hugoniot condition. This extension of the theory is particularly relevant to phase


boundaries in a van der Waals fluid. Such a fluid is described by the usual Euler equations of an 

inviscid, isentropic compressible fluid. The unknowns are the density ρ and the velocity v. The 

equations govern the conservation of mass and momentum: 

∂tρ + div(ρv) = 0, (51) 

∂t(ρv) + Div(ρv ⊗ v) +∇xp = 0. (52) 

The pressure is given as a non-monotone equation of state p = π(ρ). The function π is increasing 

on (0,ρ−) (gas phase) and (ρ+, +∞) (liquid phase), while being decreasing over (ρ−,ρ+). 

Notice that (51), (52) imply formally the conservation of energy 

 

1 

∂t 

2 ρ|v|2 

1 

+ ρε(ρ) + div 

2 ρ|v|2 

+ ρε(ρ) + π(ρ) v = 0, (53) 

where ε is the function defined by 

dε 

dρ 

π(ρ) 

= . 

ρ2 The gas and liquid phases are the states for which system (51), (52) is hyperbolic. The sound 

speed c is then the real number √ π ′ (ρ). Given a discontinuity across a smooth hypersurface, it 

fulfills the Lax shock condition if the normal velocity V of the shock front satisfies the inequalities 

(v · ν + c)+,(v· ν − c)−


that the linearized problem around a phase boundary has a variational structure too, although of a 

slightly different form than the one studied in the present article. Thus the main result of [2], the 

existence of finite energy surface waves in this linearized problem is likely to be a consequence 

of this structure, in the same spirit as in our Theorem 3.3. We leave this justification for a future 

work. 

7. General domain and variable coefficient 

We turn towards the realistic case where Ω is a smooth open domain in Rd and the quadratic 

energy density W(x;∇xu) may depend smoothly upon the space variable. For the sake of simplicity, 

we limit ourselves to bounded domains. The smoothness required for the boundary and 

the coefficients is C2 . The Lagrangian 

 

 

L[u]:= |∂tu| 2 − W(x;∇xu) dxdt 

Ω 

defines an initial boundary-value problem in Ω. From this IBVP, we define a family of IBVPs 

with constant coefficients in half-spaces, parametrized by the elements of the boundary ∂Ω.To 

each point x0 ∈ ∂Ω, we associate the Lagrangian 

 

 

L[u]:= |∂tu| 2 − W(x0;∇xu) dxdt, 

ω(x0) 

where ω(x0) is the half-space with the same outer normal ν(x0) at x0 as Ω: 

ω(x0) ={x: (x − x0) · ν(x0)


This result follows immediately from the Hille–Yosida theorem and the following estimate. 

Theorem 7.2. Under the assumptions of Theorem 7.1, there exists two positive constants ɛ and C, 

such that 

Proof. If x1 ∈ Ω, let us define 

W[u] ɛ∇xu 2 

L2 − Cu2 

(ω) L2 (ω) . 

 

Y[x1; u]:= 

R d 

W(x1;∇xu) dx. 

By rank-one convexity, there exists a positive number α(x1) such that Y[x1; u] α(x1)∇xu 2 , 

where ·stands for the L 2 (R d )-norm. Since W varies smoothly with x, and since Ω is compact, 

we have 

inf 

x1∈Ω α(x1)>0. 

Likewise, Theorem 3.5 tells that if x0 is a boundary point, then W[x0; u] dominates 

 

β(x0) ∇xu 2 dx 

ω(x0) 

for some positive number β(x0). Again, continuity and compactness of the boundary imply 

inf 

x0∈∂Ω β(x0)>0. 

In short, there exists a number γ>0 such that for every interior point x1 or boundary point x0, 

there holds true 

Y[x1; u] γ ∇xu 2 , W[x0; u] γ ∇xu 2 , 

where we employ the L2-norm, either of Rd or the half-space ω(x0), according to the context. 

Let the ball B = B(x1; r) be contained in Ω. Ifu∈ H 1 (Ω) has support contained in B, 

extension by zero yields a ũ ∈ H˙ 1 (Rd ). Then 

 

 

W[u]=Y[x1;ũ]+ W(x;∇xu) − W(x1;∇xu) dx. 

Because of smoothness, we derive 

B 

W[u] γ ∇xu 2 − O(r)∇xu 2 . 

Therefore, choosing r small enough, we are certain that 

W[u] γ 2 

∇xu 

2


holds true. We point out that the same positive r may be chosen for every point x1. To treat the 

case of a boundary point x0, we employ the same argument, but we need another technical tool. 

If r is small enough, then B ∩ Ω is diffeomorphic to a half-ball 

B+ := x: (x − x0) · ν(x0)0 as above. By compactness we may cover Ω by a finite collection of balls Bj := 

B(x j ; r) where either Bj ⊂ Ω or x j ∈ ∂Ω (here it is useful to take r less than the minimum of 

the curvature of the boundary). Let (ρ1,...,ρN) be a partition of unity over Ω adapted to the 

B ′ j s, with ρj 0. If u ∈ H 1 (Ω), we define 

Using the polar form Ψ of W ,wehave 

uj := ρj u, ujk := √ ρj ρk u.


W(x;∇u) = 

W(x;∇uj ) + 2 

Ψ(x;∇uj , ∇uk) 

j 

j


where Λ, S and Σ are Hermitian, S is linear and Σ quadratic in η and 

ξ 2 Λ(x0) − ξS(x0; η) + Σ(x0; η) α ξ 2 +|η| 2 , ∀(ξ, η) ∈ R d , ∀x0 ∈ ∂Ω, 

for some positive α (strict rank-one convexity). 

Acknowledgments 

This work was initiated by a conversation with Luc Tartar, whose fundamental questions are 

unfortunately not solved here. The idea that steady ill-posed problems have something to do with 

the transition to instability was suggested by Constantin Dafermos. I am grateful to both of them 

for their questions and their interest in the topic. Several crucial results were obtained during a 

stay in the Marais Poitevin; I thank Pascale for having let me steal hours from our nice vacations. 

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On the super fixed point property in product spaces 

Andrzej Wi´snicki 

Institute of Mathematics, Maria Curie-Skłodowska University, 20-031 Lublin, Poland 

Received 29 October 2005; accepted 10 March 2006 

Available online 2 May 2006 

Communicated by G. Pisier 

Abstract 

We prove that if F is a finite-dimensional Banach space and X has the super fixed point property for 

nonexpansive mappings, then F ⊕ X has the super fixed point property with respect to a large class of 

norms including all lp norms, 1 p

448 A. Wi´snicki / Journal of Functional Analysis 236 (2006) 447–456 

sufficient condition for X to have WFPP. (Recall that a Banach space X has normal structure if 

the Chebyshev radius rC(C) < diam C for all bounded closed convex subsets C of X consisting 

of more than one point.) 

The problem of whether FPP or WFPP is preserved under direct sum of Banach spaces is an 

old one. In 1968 Belluce, Kirk and Steiner [3] proved that the direct sum of two Banach spaces 

with normal structure, endowed with the “maximum” norm, also has normal structure. Since 

then, the preservation of normal structure and conditions which guarantee normal structure have 

been studied extensively and the problem is now quite well understood (see [28–30,35] and 

the references given there). But the situation is much more difficult if at least one of these spaces 

lacks (weak) normal structure. To the author’s knowledge the only known results are given in [10, 

19,27,30,39]. (We should also mention [5,21,23,26,37], where nonexpansive mappings defined 

on rectangles C1 × C2 are considered.) 

In this paper we prove in Section 3 that if F is finite-dimensional and X has the super fixed 

point property (see Section 2 for the definition), then F ⊕ X has the super fixed point property 

with respect to a large class of norms including all strictly monotone norms. This provides a 

solution to the “super-version” of the problem of M.A. Khamsi [21, p. 999] for this class of 

norms. Some consequences of this theorem and examples of Banach spaces with the super fixed 

point property are given in Section 4. 

2. Basic notions and tools 

In this section we shall briefly recall basic tools used in the sequel. For more details we refer 

the reader to [1,2,7,14,24,32,34]. Let X and Y be Banach spaces and let 0

A. Wi´snicki / Journal of Functional Analysis 236 (2006) 447–456 449 

Here lim U denotes the ultralimit over U. One can prove that the quotient norm on (X)U is given 

by 

 

(xn)U 

= limU xn, 

where (xn)U is the equivalence class of (xn). It is also clear that X is isometric to a subspace 

of (X)U by the embedding x → (x)U . We shall not distinguish between x and (x)U . 

The connection between finite representability and ultrapowers was independently observed 

by Henson and Moore [17], and Stern [36] (see also [1,15,34]). 

Theorem 2.2. A Banach space Y is finitely representable in X if and only if there exists an 

ultrafilter U such that Y is isometric to a subspace of (X)U . 

From Theorem 2.2 we obtain immediately 

Proposition 2.3. For any Banach space X: 

(i) X is superreflexive iff each ultrapower (X)U is reflexive; 

(ii) X has SFPP iff each ultrapower (X)U has FPP. 

We shall also need the following result concerning iterated ultrapowers, see for instance 

[34, Theorem 13.2]. 

Theorem 2.4. Let U and V be ultrafilters on I and J , respectively, and let X be a Banach 

space. Then there exists an ultrafilter W defined on I × J such that ((X)U )V is isometric to the 

ultrapower (X)W . 

Assume now that C is a nonempty weakly compact, convex subset of a Banach space X and 

that there exists a nonexpansive mapping T : C → C without fixed points. Then, by Zorn lemma, 

there exists a minimal convex and weakly compact set K ⊂ C which is invariant under T and 

which is not a singleton. Recall that a sequence (xn) is called an approximate fixed point sequence 

for T if 

lim 

n→∞ Txn − xn=0. 

The following lemma was independently proved by Goebel [13] and Karlovitz [20]. 

Lemma 2.5. If (xn) is an approximate fixed point sequence for T in K, then 

for all x ∈ K. 

lim 

n→∞ xn − x=diam K 

In 1980 the Banach space ultrapower construction was applied in fixed point theory by Maurey, 

see [31]. Let U be a free ultrafilter on N and denote by K ⊂ (X)U the set 

K = (xn)U ∈ (X)U : xn ∈ K for all n ∈ N .


We may extend the mapping T by setting T ((xn)U ) = (T xn)U . It is not difficult to see that 

T : K → K is a well-defined nonexpansive mapping. Moreover, Fix T , the set of fixed points 

of T , is nonempty and is characterized as those points from K which are represented by sequences 

(xn) in K for which lim U Txn − xn=0. We can now rephrase Lemma 2.5 as follows. 

Lemma 2.6. For every x ∈ K and y ∈ Fix T 

Let (xn) be a sequence in K and put 

x − y=diam K. 

H(xn) = (xkn )U : (xkn ) is a subsequence of (xn) . 

It is not difficult to see that if (xn) is an approximate fixed point sequence for T , then 

H(xn) ⊂ Fix T . The following lemma is a reformulation of known facts, see the arguments in 

[25, Theorem 4.4], [38, Theorem 3.1], [1, p. 86]. We sketch the proof for the convenience of the 

reader. 

Lemma 2.7. Let X be superreflexive and assume that (xn) is a sequence in K which converges 

weakly to 0. Then 

0 ∈ convH(xn). 

Proof. It is sufficient to prove that 0 is in the weak closure of H(xn). Letε>0 and let 

F1, F2,...,Fm be functionals in (X) ∗ U . Since X is superreflexive, then, by the Henson–Moore 

theorem [16,17], see also [15,34], there exist sequences (f 1 n ), (f 2 n ),...,(fm n ) ⊂ X∗ such that 

 

F1 (un)U = limU f 1 n (un), ..., 

 

Fm (un)U = limU f m n (un) 

for every (un)U ∈ X. Since (xn) converges weakly to 0, there exists an increasing sequence of 

integers (kn) such that 

for all n ∈ N. Hence 

 

f 1 n (xkn ) ε, ..., 

 

f m n (xkn ) ε 

 

F1 (xkn )U 

 

ε, ..., Fm (xkn )U 

 

ε 

and it follows that (xkn )U ∈ H(xn) belongs to the weak neighbourhood of 0. This completes the 

proof since ε>0 and F1, F2,...,Fm ∈ (X) ∗ U were arbitrary. ✷ 

3. Main theorem 

Let X and Y be Banach spaces and p ∈[1, ∞). We denote by X ⊕p Y the product space 

X ⊕ Y equipped with the norm 

 

(x, y) = x p +y p 1/p .


In [21] a larger class of norms were considered. We shall also consider another class of norms. 

A norm ·on X ⊕ Y is said to be of type (UL) if: 

(i) (x, 0)=x and (0,y)=y for every x ∈ X, y ∈ Y ; 

(ii) x (x, y) and y (x, y) for every x ∈ X, y ∈ Y ; 

(iii) there exists M>0 such that for every x,x ′ ∈ X and y ∈ Y we have 

x−x ′ M (x, y) − (x ′ ,y) . 

It was asked in [21, p. 999] whether WFPP (FPP) is preserved under the product X ⊕ Y if one of 

these spaces is finite-dimensional. 

Below we solve the “super-version” of this problem for strictly monotone norms and norms 

of type (UL). Notice that (X ⊕ Y)U is isometric to (X)U ⊕ (Y )U for any ultrafilter U and 

 

 

(xn)U ,(yn)U 

(X)U ⊕(Y )U = 

(xn,yn)U 

 

(X⊕Y)U = lim 

(xn,yn) 

U 

. X⊕Y 

From now on we shall use the same symbol ·for the norms above. 

Lemma 3.1. If the product X ⊕ Y is endowed with a norm of type (UL), then the induced space 

(X)U ⊕ (Y )U is endowed with a norm of type (UL), too. 

Proof. It is not difficult to see that 

 

(xn)U , 0 

= lim(xn, 0) 

U 

= lim xn= 

U 

(xn)U 

 

for every (xn)U ∈ (X)U and, similarly, (0,(yn)U )=(yn)U for (yn)U ∈ (Y )U . To prove (ii) 

notice that 

Hence 

lim U xn lim U 

 

(xn,yn) 

and limU yn lim (xn,yn) 

U 

. 

 

(xn)U 

(xn)U ,(yn)U 

and (yn)U (xn)U ,(yn)U 

 

for every (xn)U ∈ (X)U ,(yn)U ∈ (Y )U . Finally 

lim xn−lim x 

U U ′ 

n M lim (xn,yn) 

U 

 

− limU ′ 

(x n ,yn) 

which proves (iii). ✷ 

We are now in a position to prove the following theorem. 

Theorem 3.2. Let X be a Banach space with the super fixed point property and F be a finitedimensional 

space. Then F ⊕ X, endowed with a norm of type (UL), has the super fixed point 

property.


Proof. Assume conversely that F ⊕ X does not have SFPP. Then, by Theorem 2.2, there exists 

an ultrafilter V such that F ⊕ (X)V ∼ = (F ⊕ X)V does not have FPP. Since X has SFPP, it follows 

from Theorem 2.1 that X is superreflexive and, by Proposition 2.3, (X)V is superreflexive. Hence 

F ⊕ (X)V is superrefexive, too. To simplify notation, put Y = (X)V. 

Let T : C → C be a nonexpansive mapping defined on a bounded closed and convex subset C 

of F ⊕ Y without fixed points. Since F ⊕ Y is superreflexive, C is weakly compact and hence 

there exists a closed convex set K ⊂ C which is minimal invariant under T .Let(xn,yn) be 

an approximate fixed point sequence for T in K. We may assume that diam K = 1 and that 

((xn,yn)) converges weakly to (0, 0) ∈ K. Then, by the Goebel–Karlovitz Lemma 2.5, 

 

lim (xn,yn) = 1. 

n→∞ 

Let U be a free ultrafilter on N. Since F is finite-dimensional, (xn) converges strongly to 0. Thus 

(xkn )U = 0 for every subsequence (xkn ) of (xn) and 

H(xn,yn) := (xkn ,ykn )U : (xkn ,ykn ) is a subsequence of (xn,yn) 

By Lemma 2.7, 

= (0,ykn )U : (ykn ) is a subsequence of (yn) . 

(0, 0) ∈ convH(xn,yn) 

and, since H(xn,yn) ⊂ Fix T , there exist distinct points (0,ykn )U ,(0,yln )U ∈ H(xn,yn) such that 

 

0, 1 2ykn + 1 2yln 

 

U = 1 

ykn 2 + 1 2yln 

 

U = r 0 and put 

D = B (0,ykn )U , 1 2 d ∩ B (0,yln )U , 1 2 d ∩ B(K,r) ∩ K. 

It is easy to see that D = ∅is convex and T(D)⊂ D. We show that if (an,bn)U ∈ D, then 

(an)U = 0. Indeed, 

and thus 

Hence 

 

(ykn 

 

− yln )U 

(ykn 

and it follows from Lemma 3.1 that (an)U = 0. 

 

− bn)U 

+ (bn − yln )U 

 

 

= 

(0,ykn − bn)U 

+ (0,bn − yln )U 

 

 

 

(−an,ykn − bn)U 

+ (an,bn − yln )U 

 

 

= 

(0,ykn − yln )U 

= (ykn − yln )U 

 

 

(0,bn − yln )U 

 

= (an,bn − yln )U 

 

. 

 

0,(bn− yln )U 

= (an)U ,(bn− yln )U

Therefore, 


D1 = (un)U : (0,un)U ∈ D 

is a subset of Y which is isometric to D. 

Define T1 : D1 → D1 as 

 

T1 (un)U = PrY 

T 

(0,un)U , 

where PrY denotes the standard projection onto Y . Notice that T1 is nonexpansive. By assumption, 

X has SFPP, and it follows from Theorem 2.4 that (Y )U = ((X)V)U has FPP. Thus there 

exists (vn)U ∈ D1 such that 

 

T1 (vn)U = (vn)U 

and consequently 

T 

(0,vn)U = (0,vn)U . 

But this contradicts Lemma 2.6 because (0,vn)U ∈ D ⊂ B(K,r). ✷ 

Let us consider another, more symmetric, class of norms. A norm ·Z on R 2 is said to be 

strictly monotone if 

 

(x1,y1) Z < (x2,y2) Z 

whenever |x1| |x2|, |y1| < |y2| or |x1| < |x2|, |y1| |y2|. We say that a norm on X ⊕ Y is 

strictly monotone if 

 

(x, y) = x, y Z 

and the norm ·Z is strictly monotone. 

Strictly monotone norms do not necessarily satisfy the condition (i): 

 

(x, 0) =x and (0,y) =y for every x ∈ X, y ∈ Y. 

However, it is not difficult to see that the proof of Theorem 3.2 is also valid in this case. 

Theorem 3.3. Let X be a Banach space with the super fixed point property and F be a finitedimensional 

space. Then F ⊕ X, endowed with a strictly monotone norm, has the super fixed 

point property. 

In particular, the result holds for l p -products F ⊕p X, p ∈[1, ∞).


4. Consequences 

It is a long-standing open problem whether all superreflexive spaces have FPP (and hence 

SFPP). However, there are two important classes of spaces which have the super fixed point 

property. Recall first that 

ɛ0(X) = sup ɛ 0: δX(ɛ) = 0 , 

where δX denotes the modulus of convexity of a Banach space X. It is well known that ɛ0(X) < 2 

implies superreflexivity of X. A recent result of García Falset et al. [12] (see also [32]), states that 

if ɛ0(X) < 2, then X has FPP. In consequence, X has SFPP since ɛ0(X) = ɛ0((X)U ). Combining 

it with Theorems 3.2 and 3.3 we obtain 

Proposition 4.1. Let X be a Banach space with ɛ0(X) < 2 and let F be a finite-dimensional 

space. Then F ⊕ X, endowed with a norm of type (UL) or a strictly monotone norm, has SFPP. 

In 1997 Prus [33] introduced the notion of uniformly noncreasy spaces. A real Banach space 

X is uniformly noncreasy if for every ε>0 there is δ>0 such that if f,g ∈ SX∗ and f −g ε, 

then diam S(f,g,δ) ε, where 

S(f,g,δ) = x ∈ BX: f(x) 1 − δ ∧ g(x) 1 − δ . 

To be precise, we put diam ∅=0. It is well known that uniformly convex as well as uniformly 

smooth spaces are uniformly noncreasy. The Bynum space l2,∞ , which is l2 space endowed with 

the norm 

x2,∞ = max x + 2, x − 

2 , 

(see [4]), and the space X √ 2 , which is l2 space endowed with the norm 

x√ 2 = maxx2, √ 

2x∞ , 

(see [3]), are examples of uniformly noncreasy spaces without normal structure. It was proved 

in [33] that all uniformly noncreasy spaces are superreflexive and have SFPP. This yields 

Proposition 4.2. Let X be uniformly noncreasy and let F be a finite-dimensional space. Then 

F ⊕ X, endowed with a norm of type (UL) or a strictly monotone norm, has SFPP. 

Remark. The above result is also valid for some generalizations of uniformly noncreasy spaces 

given in [8,9,11]. 

Other examples of spaces with the super fixed point property are given by the author’s result 

[39, Theorem 2.3]. In particular, the l p -products of uniformly noncreasy spaces have SFPP. 

Banach spaces X with the property that R ⊕ X has WFPP were studied in [27]. The following 

theorem was established for the l 1 -norm but the proof works for all strictly monotone norms.


Theorem 4.3. (See [27, Theorem 1]) Let X be a Banach space which is uniformly convex in 

every direction. If Y is a Banach space such that R ⊕ Y , endowed with a strictly monotone norm, 

has WFPP, then X ⊕ Y also has WFPP. 

Theorems 3.3 and 4.3 give 

Proposition 4.4. Let X be a Banach space which is uniformly convex in every direction. If Y has 

SFPP, then X ⊕ Y , endowed with a strictly monotone norm, has WFPP. 

We conclude with the observation concerning spaces with the Schur property. The proof is 

similar to that in Section 3. 

Proposition 4.5. Let X be a Banach space with the Schur property and let Y has SFPP. Then 

X ⊕ Y , endowed with a norm of type (UL) or a strictly monotone norm, has WFPP. 

Problem. It is natural to ask whether the results of this paper are valid for the product space 

endowed with the “maximum” norm or, more generally, with a monotone norm. 

Acknowledgment 

The author thanks the referee for valuable comments on the manuscript. 

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Brown measures of sets of commuting 

operators in a type II1 factor 

Hanne Schultz 1 


Department of Mathematics and Computer Science, University of Southern Denmark, Denmark 

Received 14 November 2005; accepted 9 March 2006 



Abstract 

Using the spectral subspaces obtained in [U. Haagerup, H. Schultz, Invariant subspaces of operators in a 

general II1-factor, preprint, 2005], Brown’s results (cf. [L.G. Brown, Lidskii’s theorem in the type II case, 

in: H. Araki, E. Effros (Eds.), Geometric Methods in Operator Algebras, Kyoto, 1983, in: Pitman Res. 

Notes Math. Ser., vol. 123, Longman Sci. Tech., 1986, pp. 1–35]) on the Brown measure of an operator 

in a type II1 factor (M,τ) are generalized to finite sets of commuting operators in M. Itisshownthat 

whenever T1,...,Tn ∈ M are mutually commuting operators, there exists one and only one compactly 

supported Borel probability measure μT1,...,Tn on B(Cn ) such that for all α1,...,αn ∈ C, 

τ log |α1T1 +···+αnTn − 1| 

= 

C n 

log |α1z1 +···+αnzn − 1| dμT1,...,Tn (z1,...,zn). 

Moreover, for every polynomial q in n commuting variables, μ q(T1,...,Tn) is the push-forward measure of 

μT1,...,Tn via the map q : Cn → C. 

In addition it is shown that, as in [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general 

II1-factor, preprint, 2005], for every Borel set B ⊆ C n there is a maximal closed T1-,...,Tn-invariant subspace 

K affiliated with M, such that μT1|K,...,Tn|K is concentrated on B. Moreover, τ(P K) = μT1,...,Tn (B). 

This generalizes the main result from [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general 

II1-factor, preprint, 2005] to n-tuples of commuting operators in M. 


E-mail address: schultz@imada.sdu.dk. 

1 As a student of the PhD-school OP-ALG-TOP-GEO the author is partially supported by the Danish Research Training 

Council. Partially supported by The Danish National Research Foundation. 


doi:10.1016/j.jfa.2006.03.003

458 H. Schultz / Journal of Functional Analysis 236 (2006) 457–489 

Keywords: Brown measure; Invariant subspaces; II1-factor; Unbounded idempotents 


In [4] Brown showed that for every operator T in a type II1 factor (M,τ) there is one and 

only one compactly supported Borel probability measure μT on C, theBrown measure of T , 

such that for all λ ∈ C, 

τ log |T − λ1| 

= log |z − λ| dμT (z). 

C 

In [7] it was shown that given T ∈ M and B ∈ B(C), there is a maximal closed T -invariant 

projection P = PT (B) ∈ M, such that the Brown measure of PTP (considered as an element 

of P MP ) is concentrated on B. Moreover, PT (B) is hyper-invariant for T , 

τ PT (B) = μT (B), (1.1) 

and the Brown measure of P ⊥ TP ⊥ (considered as an element of P ⊥ MP ⊥ ) is concentrated 

on B c . 

In particular, if S,T ∈ M are commuting operators and A,B ∈ B(C), then PS(A) ∧ PT (B) is 

S- and T -invariant. Thus, it is tempting to define a Brown measure for the pair (S, T ), μS,T ,by 

μS,T (A × B) = τ PS(A) ∧ PT (B) , A,B ∈ B(C). (1.2) 

In order to show that μS,T extends (uniquely) to a Borel probability measure on C2 , we introduce 

the notion of an idempotent valued measure (cf. Section 3), and for T ∈ M and B ∈ B(C) we 

define an unbounded idempotent affiliated with M, eT (B), byD(eT (B)) = KT (B) + KT (Bc ) 

and 

 

ξ, ξ ∈ KT (B), 

eT (B)ξ = 

0, ξ ∈ KT (Bc ), 

where KT (B) is the range of PT (B) (cf. Section 3). We prove that the map B ↦→ eT (B) is an 

idempotent valued measure and this enables us to show that μS,T given by (1.2) has an extension 

to a measure on C2 (cf. Section 4). As in [7] we also prove the existence of spectral subspaces 

for a set of commuting operators in M. In the case of two commuting operators S,T ∈ M, we 

show that for B ∈ B(C2 ) there is a maximal S- and T -invariant projection P ∈ P(M), such 

that μS|P(H),T |P(H) 

(1.3) 

(computed relative to P MP ) is concentrated on B. In the case where B = 

B1 × B2, B1,B2 ∈ B(C), P is simply the intersection PS(B1) ∧ PT (B2) (cf. Section 5). 

Finally, in Section 6 we show that μS,T has a characterization similar to the one given by 

Brown in [4]. That is, we show that μS,T is the unique compactly supported Borel probability 

measure on B(C2 ) such that for all α, β ∈ C, 

τ log |αS + βT − 1| 

= log |αz + βw − 1| dμS,T (z, w), 

C 2

H. Schultz / Journal of Functional Analysis 236 (2006) 457–489 459 

and there is a similar characterization of the Brown measure of an arbitrary finite set of commuting 

operators. In particular, μαS+βT is the push-forward measure of μS,T via the map 

(z, w) ↦→ αz + βw. In fact we can show that for every polynomial q in two commuting variables, 

μq(S,T ) is the push-forward measure of μS,T via the map q : C2 → C. All this can (and 

will) be generalized to arbitrary finite sets of mutually commuting operators. 

Section 2 is devoted to a summary of [10] in which we recall the definition of the measure 

topology on M and how the closure of M with respect to the measure topology may be identified 

with the set of closed, densely defined operators affiliated with M. Afterwards we focus our 

attention on the set of unbounded idempotents affiliated with M, I( ˜M). We prove that these 

are in one-to-one correspondence with pairs of projections P,Q ∈ M with P ∧ Q = 0 and 

P ∨ Q = 1. More precisely, when E is a closed, densely defined, unbounded operator affiliated 

with M with E · E = E, then P , the range projection of E, and Q, the projection onto the kernel 

of E, satisfy that P ∧ Q = 0 and P ∨ Q = 1. Moreover, D(E) = P(H) + Q(H), and E is given 

by 

 

ξ, ξ ∈ P(H), 

Eξ = 

(1.4) 

0, ξ ∈ Q(H). 

Vice versa, when P,Q∈ M with P ∧ Q = 0 and P ∨ Q = 1, then (1.4) defines a closed, densely 

defined, unbounded operator affiliated with M with E · E = E. In particular, (1.3) defines a 

closed, densely defined idempotent affiliated with M. In Section 2 it is also shown that I( ˜M) is 

stable with respect to addition of countably many idempotents (En) ∞ n=1 with EnEm = EmEn = 0, 

n = m. These are results which are needed in the succeeding sections. 

2. Idempotents in ˜M 

We begin this section with a summary of E. Nelson’s “Notes on non-commutative integration” 

[10]. We consider a finite von Neumann algebra M represented on a Hilbert space H and 

we fix a faithful, normal, tracial state τ on M.WeletP(M) denote the set of projections in M. 

Nelson defines the measure topology on M as follows. For ε, δ > 0, let 

N(ε, δ) = T ∈ M |∃P ∈ P(M): TP ε, τ P ⊥ δ . 

The measure topology on M is then the translation invariant topology on M for which the 

N(ε, δ)’s form a fundamental system of neighborhoods of 0. ˜M is the completion of M with 

respect to the measure topology. 

Similarly, Nelson defines ˜H to be the completion of H with respect to the translation invariant 

topology on H for which the sets 

O(ε, δ) = ξ ∈ H |∃P ∈ P(M): Pξ ε, τ P ⊥ δ 

form a fundamental system of neighborhoods of 0. According to [10, Theorem 2], the natural 

mappings M → ˜M and H → ˜H are both injections. 

Theorem 2.1. [10, Theorem 1] The mappings 

M → M : T ↦→ T ∗ , M × M → M : (S, T ) ↦→ S + T, M × M → M : (S, T ) ↦→ S · T, 

H × H → H : (ξ, η) ↦→ ξ + η, M × H → H : (T , ξ) ↦→ Tξ


all have unique continuous extensions as mappings ˜M → ˜M, ˜M × ˜M → ˜M, ˜M × ˜M → ˜M, 

˜H × ˜H → ˜H and ˜M × ˜H → ˜H, respectively. In particular, ˜H is a complex vector space and ˜M 

is a complex ∗-algebra with a continuous representation on ˜H. 

For x ∈ ˜M define 

and define Mx : D(Mx) → H by 

D(Mx) ={ξ ∈ H | xξ ∈ H} (2.1) 

Mxξ = xξ, ξ ∈ D(Mx). (2.2) 

Recall that a (not necessarily bounded or everywhere defined) operator A on H is said to be 

affiliated with M if AU = UA for every unitary U ∈ M ′ . 

Theorem 2.2. [10, Theorem 4] For every x ∈ ˜M, Mx is a closed, densely defined operator 

affiliated with M, and 

Moreover, for x,y ∈ ˜M, 

M ∗ x 

where A denotes the closure of a closable operator A. 

= Mx ∗. (2.3) 

Mx+y = Mx + My, (2.4) 

Mx·y = Mx · My, (2.5) 

A closed, densely defined operator A on H has a polar decomposition A = V |A|, and if A is 

affiliated with M, then V ∈ M and all the spectral projections (E|A|([0,t[))t>0 belong to M. 

Put 

An = V 

n 

0 

t dE|A|(t). 

Assuming that A is affiliated with M, we get that (An) ∞ n=1 is a Cauchy sequence with respect to 

the measure topology. Indeed, 

 

(An+k − An) · E|A| [0,n[ = 0, 

and τ(E|A|([n, ∞[)) → 0asn →∞. Hence, there exists a ∈ ˜M such that An → a in measure, 

and according to [10, Theorem 3], A = Ma. It follows that every closed, densely defined operator 

A affiliated with M is of the form A = Ma for some a ∈ ˜M, and this a is uniquely determined. 

Indeed, if Ma = Mb, then a and b agree on D(Ma) which is dense in H with respect to the norm 

topology and hence dense in ˜H with respect to the measure topology. Since the representation of 

˜M on H is continuous, it follows that a and b agree on all of ˜H. By the same argument, if S and


T are closed, densely defined operators affiliated with M which agree on a dense subset of H, 

then S = T . 

In summary, ˜M is the completion of M with respect to the measure topology but it may also 

be viewed as the set of closed, densely defined operators affiliated with M. In particular, if S and 

T belong to the latter, then Theorem 2.2 tells us that S ∗ , S + T and S · T are also closed, densely 

defined and affiliated with M. 

Notation 2.3. In what follows we shall identify ˜M with the set of closed, densely defined operators 

affiliated with M, but whenever necessary, we will specify which one of the two pictures 

mentioned above we are using. In general, lower case letters will represent elements of the completion 

of M with respect to the measure topology, whereas upper case letters will represent 

closed, densely defined operators affiliated with M. Forx ∈ ˜M we will denote by ker(x), 

range(x) and supp(x) the kernel of Mx, the range of Mx and the support projection of Mx, 

respectively. 

In the rest of this section we will study the set of idempotent elements in ˜M, 

I( ˜M) ={e ∈ ˜M | e · e = e}. (2.6) 

Alternatively, if ˜M is viewed as the set of closed, densely defined operators affiliated with M, 

then I( ˜M) is the subset of operators E fulfilling that E · E = E. 

The following proposition shows that I( ˜M) is in one-to-one correspondence with the set of 

pairs of projections P,Q∈ M such that P ∧ Q = 0 and P ∨ Q = 1. 

Proposition 2.4. Let E ∈ I( ˜M). Then the range of E, range(E), and the kernel of E, ker(E) 

(which is the range of 1 − E), are closed subspaces of H, with 

and 

range(E) ∩ ker(E) ={0} (2.7) 

range(E) + ker(E) = H. (2.8) 

Moreover, D(E) = range(E)+ker(E). Conversely, if P , Q ∈ M are projections with P ∧Q = 0 

and P ∨ Q = 1, then there is a unique idempotent E ∈ ˜M with D(E) = P(H) + Q(H), 

range(E) = P(H) and ker(E) = Q(H), and E is determined by 

 

ξ, ξ ∈ P(H), 

Eξ = 

(2.9) 

0, ξ ∈ Q(H). 

Proof. Write E = Me for a (uniquely determined) element e ∈ ˜M with e · e = e and recall 

from (2.1) that 

D(E) ={ξ ∈ H | eξ ∈ H}. 

Let E · E denote the densely defined operator on H obtained by composing E with itself. Then 

D(E · E) = ξ ∈ D(E) | Eξ ∈ D(E) = ξ ∈ D(E) | eEξ ∈ H = ξ ∈ D(E) | e 2 ξ ∈ H .


But e 2 = e as everywhere defined operators on ˜H. Hence, for ξ ∈ D(E), 

e 2 ξ = eξ = Eξ, 

and it follows that D(E · E) = D(E) and that E · E = E (without taking closure). In particular, 

range(E) ⊆ D(E). Moreover, since E · E = E, 

range(E) = ξ ∈ D(E) | Eξ = ξ = ker(1 − E). 

According to [8, Exercise 2.8.45], the kernel of a closed operator is closed. Hence, ker(E) and 

range(E) = ker(1 − E) are closed. Moreover, 

range(E) ∩ ker(1 − E) = ker(1 − E) ∩ ker(E) ={0}. 

Clearly, range(E) + ker(E) ⊆ D(E), and since 

the converse inclusion also holds. That is, 

ξ = Eξ + (1 − E)ξ, ξ ∈ D(E), 

D(E) = range(E) + ker(E). 

Let P and Q denote the projections onto range(E) and ker(E), respectively. Then by the above, 

P ∧ Q = 0 and P ∨ Q = 1, and E is the operator on D(E) = P(H) + Q(H) determined by (2.9). 

In particular, E is uniquely determined by its range and its kernel. 

Conversely, assume that P and Q are projections in M, such that P ∧ Q = 0 and P ∨ Q = 1, 

and let E be the operator on D(E) = P(H)+Q(H) determined by (2.9). Then clearly, E ·E = E 

(without taking closure), range(E) = P(H), and ker(E) = Q(H). Moreover, the graph of E is 

given by 

G(E) = (ξ + η,ξ) | ξ ∈ P(H), η ∈ Q(H) 

= (u, v) ∈ H × H | u − v ∈ Q(H), v ∈ P(H) , 

which is clearly a closed subspace of H × H. Hence, E ∈ I( ˜M). ✷ 

Definition 2.5. We define tr : I( ˜M) →[0, 1] by 

where supp(e) ∈ P(M) denotes the support projection of Me. 

tr(e) = τ supp(e) , e∈ I( ˜M), (2.10) 

Remark 2.6. For every x ∈ ˜M, supp(x) ∼ Prange(x) so one also has that for e ∈ I( ˜M), 

tr(e) = τ(Prange(e)). (2.11) 

Throughout the paper we will, without further mentioning, make use of this identity.


Proposition 2.7. Let e1,...,en ∈ I( ˜M) (seen as everywhere defined operators on ˜H) with 

eiej = 0 when i = j. Then e1 +···+en ∈ I( ˜M), and 

(a) ker(e1 +···+en) = n i=1 ker(ei); 

(b) supp(e1 +···+en) = n i=1 supp(ei); 

(c) range(e1 +···+en) = n i=1 range(ei); 

(d) tr(e1 +···+en) = n i=1 tr(ei). 

Proof. It suffices to consider the case n = 2. The general case follows by induction over n ∈ N. 

If e1,e2 ∈ I( ˜M) with e1e2 = e2e1 = 0, then obviously, e1 + e2 ∈ I( ˜M). 

(a) Clearly, ker(e1) ∩ ker(e2) ⊆ ker(e1 + e2). On the other hand, if ξ ∈ ker(e1 + e2), then 

eiξ = ei(e1 + e2)ξ = 0(i = 1, 2), whence ker(e1 + e2) ⊆ ker(e1) ∩ ker(e2). 

(b) Since supp(e1 + e2) is the projection onto [ker(e1 + e2)] ⊥ , (b) follows from (a). 

(c) For a closed, densely defined operator S affiliated with M, range(S) = ker(S∗ ) ⊥ .Using 

that every idempotent has closed range (cf. Proposition 2.4) and applying (a) to e∗ 1 +···+e∗ n ,we 

find that 

range(e1 +···+en) = range(e1 +···+en) = ker e ∗ 1 +···+e∗ ⊥ n 

= 

n 

i=1 

ker e ∗⊥ i = 

n 

range(ei). 

(d) For i = 1, 2 put Pi = Prange(ei). Since e1e2 = e2e1 = 0, for ξ ∈ P1(H) ∩ P2(H) we have 

that 

i=1 

ξ = e1ξ = e2e1ξ = 0. 

Then by Kaplansky’s formula (cf. [8, Theorem 6.1.7]), 

so that 

P1 ∨ P2 − P1 ∼ P2 − P1 ∧ P2 = P2, 

τ(P1 ∨ P2) = τ(P1) + τ(P2). 

According to (c), P1 ∨ P2 is the projection onto range(e1 + e2), and then by Remark 2.6, 

tr(e1 + e2) = tr(e1) + tr(e2). ✷ 

Proposition 2.8. Let (en) ∞ n=1 be a sequence of idempotents in ˜M with enem = emen = 0 when 

n = m. Then there is an idempotent in ˜M, which we denote by ∞ n=1 en, such that N n=1 en → 

∞n=1 en in measure as N →∞. Moreover, supp( N n=1 en) ↗ supp( ∞ n=1 en), whence 

 

∞ 

supp 

 

∞ 

= supp(en), (2.12) 

n=1 

en 

n=1


and 

Also, 

Proof. Let 

range 

 

∞ 

tr 

n=1 

∞ 

n=1 

en 

en 

fn = 

k=1 

 

∞ 

= tr(en). (2.13) 

n=1 

 

∞ 

= range(en). (2.14) 

n=1 

n 

ek, n∈N. (2.15) 

Then, according to Proposition 2.7, supp(fn) = n k=1 supp(ek), and tr(fn) = n k=1 tr(ek). We 

prove that (fn) ∞ n=1 is a Cauchy sequence in ˜M.Forn, k ∈ N,let 

Then 

so for every ε>0, 

Pn,k = supp(fn+k − fn) = 

(fn+k − fn)P ⊥ n,k 

k 

supp(en+l). (2.16) 

l=1 

fn+k − fn ∈ N ε, τ(Pn,k) 

= N ε, 

= 0, (2.17) 

k 

 

tr(en+l) . (2.18) 

Now, n k=1 tr(ek) = tr(fn) 1, so for arbitrary δ>0 there is an n0 ∈ N such that 

l=1 

∞ 

tr(ek) δ. (2.19) 

k=n0 

It follows from (2.18) and (2.19) that when n n0 and k 1, then fn+k − fn ∈ N(ε, δ). Thus, 

(fn) ∞ n=1 is a Cauchy sequence in ˜M. Put 

For all k,n ∈ N, fn+kfn = fnfn+k = fn, and hence 

so that e · e = e. 

e = lim 

n→∞ fn ∈ ˜M. (2.20) 

efn = fne = fn, (2.21)


Let Pn = supp(fn) = n k=1 supp(ek). Then Pn ↗ P := ∞ k=1 supp(ek) as n →∞, and 

∞ 

tr(ek) = lim 

k=1 

n 

n→∞ 

k=1 

tr(ek) = lim 

n→∞ tr(fn) = lim 

n→∞ τ(Pn) = tr(P ). 

It follows from (2.21) that for every n ∈ N, Pn supp(e). Hence P supp(e). On the other 

hand, for every n ∈ N, fn(1 − Pn) = 0, so 

 

e(1 − P)= lim fn(1 − Pn) 

n→∞ 

= 0 (2.22) 

(the limit refers to the measure topology). Thus, supp(e) P , and we have shown that supp(e) = 

P = ∞ k=1 supp(ek) and that (2.13) holds. 

In order to prove (2.14), let ξ ∈ range(em). Then emξ = ξ and 

Thus, 

range 

∞ 

en 

n=1 

∞ 

n=1 

 

ξ = emξ = ξ. 

en 

 

∞ 

⊇ range(en). (2.23) 

On the other hand, we know that range( ∞ n=1 en) ⊆ H, and since range( N n=1 en) ⊆ 

∞n=1 range(en) for all N ∈ N,wealsohavethat 

range 

∞ 

n=1 

en 

n=1 

 

∞ 

⊆ range(en) 

where ∼ denotes closure with respect to the measure topology. Intersecting by H on both sides 

of the inclusion, we get that 

This proves (2.14). ✷ 

range 

∞ 

n=1 

en 

n=1 

∼ 

, 

 

∞ 

⊆ range(en). (2.24) 

We shall make use of the following theorem from [2]. For a published proof of it, we refer the 

reader to [1]. 

n=1


Theorem 2.9. [2] Let E and F be (not necessarily closed) subspaces of H which are affiliated 

with M. 2 Then E ∩ F is affiliated with A, and 

E ∩ F = E ∩ F. (2.25) 

Lemma 2.10. Consider idempotents e, f ∈ ˜M. Let P = Prange(e), Q = Prange(1−e), R = Prange(f ) 

and S = Prange(1−f). Then ef = feif and only if 

Proof. Clearly, 

(P ∧ R) ∨ (P ∧ S) ∨ (Q ∧ R) ∨ (Q ∧ S) = 1. (2.26) 

1 = ef + e(1 − f)+ (1 − e)f + (1 − e)(1 − f). (2.27) 

Suppose that ef = fe, and let g1 = ef , g2 = e(1 − f), g3 = (1 − e)f and g4 = (1 − e)(1 − f). 

Then g1,...,g4 are idempotents with support projections P1, P2, P3 and P4, respectively, such 

that 

 

4 

 

(H) = H. (2.28) 

Pi 

i=1 

Moreover, P1 P ∧ R, P2 P ∧ S, P3 Q ∧ R and P4 Q ∧ S. This shows that (2.26) holds. 

On the other hand, assume that (2.26) holds. According to (2.26) and Theorem 2.9, 

H0 := D(ef ) ∩ D(f e) ∩ (P ∧ R)(H) + (P ∧ S)(H) + (Q ∧ R)(H) + (Q ∧ S)(H) 

is dense in H so it suffices to prove that ef and fe agree on H0. To see this, let ξ ∈ D(ef ) ∩ 

D(f e) ∩ (P ∧ R)(H). Then 

ef ξ = eξ = ξ = fξ = feξ, (2.29) 

and similarly, when ξ ∈ D(ef ) ∩ D(f e) ∩ (P ∧ S)(H), ξ ∈ D(ef ) ∩ D(f e) ∩ (Q ∧ R)(H) or 

ξ ∈ D(ef ) ∩ D(f e) ∩ (Q ∧ S)(H). Thus, ef agrees with feon H0. ✷ 

3. An idempotent valued measure associated with T ∈ M 

As in the previous section, consider a finite von Neumann algebra M with a faithful, normal, 

tracial state τ . Inspired by the notion of a spectral measure we make the following definition. 

Definition 3.1. Let (X, F) denote a measurable space. An idempotent valued measure on (X, F) 

(with values in ˜M) isamape from F into I( ˜M) such that: 

(i) e(X) = 1, 

(ii) e(F1)e(F2) = e(F2)e(F1) = 0 when F1,F2 ∈ F with F1 ∩ F2 =∅, 

2 A subspace E of H is said to be affiliated with M if for all T ∈ M ′ , T(E)⊆ E. Note that if E is affiliated with M, 

then the projection onto E belongs to M,andifE is closed, then this is a necessary and sufficient condition for E to be 

affiliated with M.


(iii) when (Fn) ∞ n=1 is a sequence of mutually disjoint sets from F, then N n=1 e(Fn) converges 

in measure as N →∞to e( ∞ n=1 Fn), i.e. 

 

∞ 

 

∞ 

e = e(Fn). 

n=1 

Fn 

Note that because of (ii) and Proposition 2.8, the limit in (iii) actually exists. 

From now on we will assume that M is in fact a type II1 factor. Recall from [7] that for 

T ∈ M and B ⊆ C a Borel set there is a maximal T -invariant projection P = PT (B) ∈ M, such 

that the Brown measure of PTP (considered as an element of P MP ) is concentrated on B. 

Moreover, PT (B) is hyper-invariant for T , 

n=1 

τ PT (B) = μT (B), (3.1) 

and the Brown measure of P ⊥ TP ⊥ (considered as an element of P ⊥ MP ⊥ ) is concentrated 

on B c .WeletKT (B) denote the range of PT (B). Then the aim of this section is to prove: 

Theorem 3.2. Let T ∈ M, and for B ∈ B(C), leteT (B) with D(eT (B)) = KT (B) + KT (B c ) be 

given by 

 

ξ, ξ ∈ KT (B), 

eT (B)ξ = 

0, ξ ∈ KT (Bc ). 

Then eT (B) ∈ I( ˜M), and B ↦→ eT (B) is an idempotent valued measure. 

The proof of this theorem uses various results which we state and prove below. The first one 

of these is a lemma which we proved in [7], but for the sake of completeness we give the proof 

here as well. 

Lemma 3.3. Let T ∈ M, and let P ∈ M be a non-zero, T -invariant projection. Then for every 

B ∈ B(C), 

(3.2) 

KT |P(H) (B) = KT (B) ∩ P(H), (3.3) 

where T |P(H) is considered as an element of the type II1 factor P MP . 

Proof. Let Q ∈ P MP denote the projection onto KT |P(H) (B), and let R = PT (B) ∧ P . We will 

prove that Q R and R Q. 

Clearly, Q P . In order to see that Q PT (B), recall that PT (B) is maximal with respect 

to the properties: 

(i) PT (B)T PT (B) = TPT (B); 

(ii) μPT (B)T PT (B) (computed relative to PT (B)MPT (B)) is concentrated on B. 

Since 

QT Q = QT P Q = TPQ= TQ, (3.4)


and μQT Q = μQT P Q (computed relative to QMQ) is concentrated on B, we get that Q 

PT (B), and hence Q R. 

Similarly, to prove that R Q, we must show that: 

(i ′ ) RT PR = TPR,i.e.RT R = TR; 

(ii ′ ) μRT PR = μRT R (computed relative to RMR) is concentrated on B. 

Note that if PT (B) = 0, then R Q, so we may assume that PT (B) = 0. (i ′ ) holds, because 

R(H) = P(H) ∩ PT (B)(H) is T -invariant when P(H) and PT (B)(H) are T -invariant. In order 

to prove (ii ′ ), at first note that R(H) is TPT (B)-invariant. Hence 

⊥ 

μTPT (B) = τ1(R) · μRT R + τ1 R · μR⊥TR⊥, (3.5) 

where 

It follows that 

τ1 = 

1 

τ(PT (B)) · τ|PT (B)MPT (B). 

c 

τ1(R) · μRT R B c 

μTPT (B) B = 0, (3.6) 

and thus, if R = 0, then μRT R(B c ) = 0, and (ii ′ ) holds. If R = 0, then R Q is trivially fulfilled. 

✷ 

Proposition 3.4. For every Borel set B ⊆ C, 

KT (B) = KT ∗ 

c 

B ∗⊥, (3.7) 

where A ∗ := {z | z ∈ A} for A ⊆ C. Moreover, for all Borel sets A,B ⊆ C, 

and 

KT (A) ∩ KT (B) = KT (A ∩ B), (3.8) 

KT (A ∪ B) = KT (A) + KT (B). (3.9) 

Proof. Let B ∈ B(C) and let P = PT (B). Then P ⊥ is T ∗-invariant, and 

∗ 

μP ⊥T ∗P ⊥ B = μ (P ⊥T ∗P ⊥ ) ∗(B) = μP ⊥TP⊥(B) = 0 (3.10) 

(recall that μP ⊥TP⊥ is concentrated on Bc ). Thus, μP ⊥T ∗P ⊥ is concentrated on C \ B∗ , and 

maximality of PT ∗(C \ B∗ ) implies that 

PT (B) ⊥ = P ⊥ PT ∗ 

∗ 

C \ B . (3.11) 

Since


τ PT ∗ 

∗ 

C \ B = μT ∗ 

∗ 

C \ B = 1 − μT ∗ 

∗ 

B = 1 − μT (B) = τ PT (B) ⊥ , 

we get from (3.11) that PT (B) ⊥ = PT ∗(C \ B∗ ). 

Next, let A,B ∈ B(C). By maximality of PT (A) and PT (B), PT (A ∩ B) PT (A) ∧ PT (B), 

so ⊇ holds in (3.8). We let K := KT (A) ∩ KT (B), and we let Q denote the projection onto K. 

Then, according to Lemma 3.3, 

K = KT |K T (A) (B) = KT |K T (B) (A), 

proving that μQT Q is concentrated on A and on B and therefore on A ∩ B. Consequently, Q 

PT (A ∩ B),so⊆ also holds in (3.8). 

Finally, we infer from (3.7) and (3.8) that 

c c 

KT (A ∪ B) = KT A ∩ B c = KT ∗ 

c c 

A ∩ B ∗⊥ = KT ∗ 

c 

A ∗ c 

∩ B ∗⊥ = KT ∗ 

c 

A ∗ ∩ KT ∗ 

c 

B ∗⊥ = KT ∗ 

 

Ac ∗⊥ + KT ∗ 

 

Bc ∗⊥ = KT (A) + KT (B). ✷ 

It follows from Propositions 3.4 and 2.4 that for B ∈ B(C), eT (B) given by (3.2) belongs to 

I( ˜M), as stated in Theorem 3.2. 

Lemma 3.5. Let (xn) ∞ n=1 be a sequence in ˜M, and suppose τ(supp(xn)) → 0 as n →∞. Then 

xn → 0 in the measure topology. 

Proof. This is standard. ✷ 

If S ∈ M commutes with T ∈ M, then for every B ∈ B(C), KT (B) and KT (B c ) are 

S-invariant, and therefore S commutes with eT (B) as well. We prove that, as a consequence 

of this, [eS(A), eT (B)]=0 for every A ∈ B(C). 

Lemma 3.6. Let T ∈ M, and let e ∈ I( ˜M) with [e,T ]=0. Then for every B ∈ B(C), 

[e,eT (B)]=0. In particular, if S ∈ M commutes with T , then [eS(·), eT (·)]=0. 

Proof. Let P = Prange(e), Q = Prange(1−e), R = Prange(eT (B)) and S = Prange(1−eT (B)). We prove 

that (2.26) holds. Since eT = Te, P(H) and Q(H) are T -invariant. Then by Lemma 3.3, 

and 

KT |P(H) (B) = KT (B) ∩ P(H) = R(H) ∩ P(H), 

c 

KT |P(H) B c 

= KT B ∩ P(H) = S(H) ∩ P(H). 

Hence (R ∧ P)∨ (S ∧ P)= P , and similarly, (R ∧ Q) ∨ (S ∧ Q) = Q. It follows that 


1 = P ∨ Q = (R ∧ P)∨ (S ∧ P)∨ (R ∧ Q) ∨ (S ∧ Q),


Proof of Theorem 3.2. eT (∅) = 0, because PT (∅) = 0. If B1,B2 ∈ B(C) with B1 ∩ B2 =∅, then 

KT (B1) ∩ KT (B2) ={0}, i.e. range(eT (B1)) ∩ range(eT (B2)) ={0}. According to Lemma 3.6, 

[eT (B1), eT (B2)]=0 so that eT (B1)eT (B2) ∈ I( ˜M). Moreover, 

range eT (B1)eT (B2) ⊆ range eT (B1) ∩ range eT (B2) ={0}, 

and we conclude that eT (B1)eT (B2) = eT (B2)eT (B1) = 0. 

Now, let (Bn) ∞ n=1 be a sequence of mutually disjoint Borel sets. Then for each N ∈ N we get 

from Proposition 3.4 and Lemma 2.7 that 

 

N 

 

N 

 

range eT Bn = KT Bn = KT (B1) +···+KT (BN ) 

n=1 

n=1 

= range eT (B1) +···+eT (BN ) 

and 

ker 

 

eT 

N 

Bn 

n=1 

 

N 

= KT 

n=1 

Bn 

c 

N 

= KT 

n=1 

= ker eT (B1) +···+eT (BN ) . 

B c n 

 

N 

= 

c 

KT Bn = 

n=1 

n=1 

N 

ker eT (Bn) 

Since an element e in I( ˜M) is uniquely determined by its kernel and its range, it follows that eT 

is additive, i.e. 

 

N 

 

= eT (B1) +···+eT (BN ) (N ∈ N). (3.12) 

n=1 

eT 

n=1 

Bn 

Additivity of eT implies that 

 

∞ 

 

∞ 

 

eT Bn − eT (Bn) = lim 

N→∞ 

n=1 

eT 

= lim 

N→∞ eT 

(the limits refer to the measure topology), where 

 

τ supp 

 

∞ 

 

= τ 

 

∞ 

eT 

as N →∞. 

Bn 

n=N+1 

PT 

∞ 

Bn 

n=1 

∞ 

Bn 

n=N+1 

n=N+1 

 

 

N 

− 

Bn 

 

= μT 

eT (Bn) 

n=1 

∞ 

n=N+1 

Bn 

 

 

→ 0, 

Combining this with (3.13) and Lemma 3.5, we find that eT is σ -additive as well. ✷ 

(3.13) 

Note that in the case where T is a normal operator, B ↦→ PT (B) is just the spectral measure 

of T , and eT (B) = PT (B).


4. The Brown measure of a set of commuting operators in M 

As in the previous section, let M be a type II1 factor. The purpose of this section is to prove: 

Theorem 4.1. Let n ∈ N, and let T1,...,Tn ∈ M be commuting operators. Then there is a probability 

measure μT1,...,Tn on (Cn , B(C n )), which is uniquely determined by 

μT1,...,Tn (B1 ×···×Bn) = τ 

n 

i=1 

PTi (Bi) 

where PTi (Bi) ∈ M is the projection onto KTi (Bi) (cf. Section 2). 

 

, B1,...,Bn ∈ B(C), (4.1) 

The idea of proof is as follows. As mentioned in the previous section (cf. Lemma 3.6), if 

S ∈ M commutes with T ∈ M, then [eS(A), eT (B)]=0 for all A,B ∈ B(C). We may therefore 

define a map eT1,...,Tn from B(C)n into I( ˜M) by 

eT1,...,Tn (B1,...,Bn) = eT1 (B1)eT2 (B2) ···eTn (Bn), B1,...,Bn ∈ B(C). (4.2) 

We will then define ν on B(C) n by 

ν(B1,...,Bn) = τ supp eT1,...,Tn (B1,...,Bn) = τ(Prange(eT1 ,...,Tn (B1,...,Bn))) 

 

n 

= τ PTi (Bi) 

 

, B1,...,Bn ∈ B(C), (4.3) 

i=1 

and we will prove that ν extends (uniquely) to a probability measure, μT1,...,Tn on (Cn , B(C n )). 

Theorem 4.2. Consider uncountable, complete, separable metric spaces (X1,d1),...,(Xn,dn). 

Suppose ν : B(X1) ×···×B(Xn) →[0, ∞[ is a map satisfying: 

(1) for all B2 ∈ B(X2), B3 ∈ B(X3),...,Bn ∈ B(Xn), B ↦→ ν(B,B2,...,Bn) is a measure on 

(X1, B(X1)); 

(2) for all B1 ∈ B(X1), B3 ∈ B(X3),...,Bn ∈ B(Xn), B ↦→ ν(B1,B,B3,...,Bn) is a measure 

on (X2, B(X2)); 

. 

(n) for all B1 ∈ B(X1), B2 ∈ B(X2), . . . , Bn−1 ∈ B(Xn−1), B ↦→ ν(B1,B2,...,Bn−1,B) is a 

measure on (Xn, B(Xn)). 

Then there is a unique measure μ on n i=1 B(Xi), such that for all B1 ∈ B(X1), B2 ∈ 

B(X2), . . . , Bn ∈ B(Xn), 

μ(B1 × B2 ×···×Bn) = ν(B1,B2,...,Bn). (4.4) 

Proof. According to [9, Remark 1, p. 358], (Xi, B(Xi)) is Borel equivalent to ([0, 1], B([0, 1])), 

i.e. there is a bijective bimeasurable map φi : (Xi, B(Xi)) → ([0, 1], B([0, 1])). Therefore we


may as well assume that Xi = R (i = 1,...,n). We may also assume that ν(R, R,...,R) = 1. 

We define F : R n →[0, 1] by 

F(x1,...,xn) = ν ]−∞,x1],...,]−∞,xn] , x1,...,xn ∈ R. (4.5) 

Because of (1)–(n), F is increasing in each variable separately and satisfies: 

(a) if x (k) 

i ↘ xi, i = 1,...,n, then F(x (k) 

1 ,...,x(k) n ) ↘ F(x1,...,xn), 

(b) if xi ↘−∞for some i ∈{1,...,n}, then F(x1,...,xn) ↘ 0, 

(c) if xi ↗∞for all i ∈{1,...,n}, then F(x1,...,xn) ↗ 1. 

Then, according to [3, Corollary 2.27], there is a (unique) probability measure μ on (R n , B(R n )) 

such that for all x1,...,xn ∈ R, 

and 

μ ]−∞,x1]×···×]−∞,xn] = F(x1,...,xn). (4.6) 

Let x2,...,xn ∈ R be fixed but arbitrary. Then the (finite) measures 

B ↦→ μ B ×]−∞,x2]×···×]−∞,xn] 

B ↦→ ν B,]−∞,x2],...,]−∞,xn] 

have the same distribution functions. Hence they must be identical. That is, for all B ∈ B(R), 

μ B ×]−∞,x2]×···×]−∞,xn] = ν B,]−∞,x2],...,]−∞,xn] . (4.7) 

Now, let B1 ∈ B(R) and x3,...,xn ∈ R be fixed but arbitrary. Then (4.7) shows that the (finite) 

measures 

B ↦→ μ B1 × B ×]−∞,x3]×···×]−∞,xn] 

and 

B ↦→ ν B1,B,]−∞,x3],...,]−∞,xn] 

have the same distribution functions, so they must be identical as well. That is, for all B ∈ B(R), 

μ B1 × B ×]−∞,x3]×···×]−∞,xn] = ν B1,B,]−∞,x3],...,]−∞,xn] . (4.8) 

Continuing like this we find that (4.4) holds. ✷ 

It follows from Theorem 4.2 that in order to show that μT1,...,Tn exists, we must prove that 

(1)–(n) of Theorem 4.2 hold in the case where X1 =···=Xn = C, and where ν is given by (4.3). 

From now on we will, in order to simplify notation a little, consider the case n = 2, and we 

will assume that S,T ∈ M are commuting operators. It should be clear that the proof given 

below may be generalized to the case of arbitrary n ∈ N.


Lemma 4.3. For fixed A ∈ B(C), νA : B(C) →[0, 1] given by 

(cf. (4.3)) is a measure on (C, B(C)). 

νA(B) = ν(A,B) = τ PS(A) ∧ PT (B) , B ∈ B(C) (4.9) 

Proof. According to Theorem 3.2 and Definition 3.1, eT (∅) = 0, so 

νA(∅) = τ supp eS(A)eT (∅) = 0. 

Let (Bn) ∞ n=1 be a sequence of mutually disjoint sets from B(C). Then eT ( ∞ n=1 Bn) = 

∞n=1 

eT (Bn),so 

 

∞ 

 

∞ 

eS(A)eT Bn = eS(A)eT (Bn) (4.10) 

n=1 

with eS(A)eT (Bn)eS(A)eT (Bm) = eS(A)eT (Bn)eT (Bm) = 0 when n = m. Hence, by Proposition 

2.8, 

 

∞ 

 

∞ 

tr eS(A)eT Bn = tr eS(A)eT (Bn) . (4.11) 

This shows that νA is a measure. ✷ 

n=1 

It now follows from Lemma 4.3 and Theorem 4.2 that there is one and only one (probability) 

measure μS,T on B(C 2 ) such that for all A,B ∈ B(C), 

n=1 

n=1 

μS,T (A × B) = τ supp eS,T (A, B) = τ PS(A) ∧ PT (B) , (4.12) 

and this proves Theorem 4.1 in the case n = 2. 

5. Spectral subspaces for commuting operators S,T ∈ M 

Theorem 5.1. Let S, T ∈ M be commuting operators, and let B ⊆ C 2 be any Borel set. Then 

there is a maximal, closed, S- and T -invariant subspace K = KS,T (B) affiliated with M, such 

that the Brown measure μS|K,T |K is concentrated on B. Let PS,T (B) ∈ M denote the projection 

onto KS,T (B). Then more precisely: 

(i) if B = B1 × B2 with B1,B2 ∈ B(C), then 

(ii) if B is a disjoint union of the sets (B (k) 

1 

then 

PS,T (B) = PS(B1) ∧ PT (B2); (5.1) 

PS,T (B) = 

∞ 

k=1 

× B(k) 

2 )∞ k=1 , where B(k) 

i ∈ B(C), k ∈ N, i = 1, 2, 

PS 

(k) (k) 

B 1 ∧ PT B 2 ; (5.2)


(iii) and for general B ∈ B(C 2 ), 

Moreover, 

PS,T (B) = 

 

B⊆U,U⊆C 2 open 

PS,T (U). (5.3) 

μS,T (B) = τ PS,T (B) , B ∈ B C 2 . (5.4) 

Remark 5.2. Every non-empty, open subset of C 2 ∼ = R 4 is a disjoint union of countably many 

standard intervals, i.e.setsoftheform 4 i=1 ]ai,bi], where −∞


μS|K,T |K 

(B) = 

= 

∞ 

k=1 

∞ 

k=1 

 

τP MP PS 

 

τP MP PS 

(k) 

B 1 ∧ PT 

(k) 

B 1 ∧ PT 

(k) 

B 2 ∧ P 

(k) 

B 2 

= 1 

∞ 

tr 

τ(P) 

k=1 

(k) (k) 

eS B 1 

eT B 2 

= 1 

τ(P) tr 

 

∞ 

(k) (k) 

eS B 1 

eT B 2 

k=1 

 

= 1 

τ(P) τ 

 

∞ 

(k) (k) 

PS B 1 ∧ PT B 2 

 

= 1. 

k=1 

Thus, (b) holds. 

Now, suppose that Q ∈ M is an S- and T -invariant projection, and that μS|L,T |L is concentrated 

on B, where L = Q(H). Then by Lemma 3.3 and Proposition 2.8, 

P ∧ Q = 

= 

∞ 

∞ 

k=1 

k=1 

PS 

PS|L 

Hence, Proposition 2.8 and (5.6) imply that 

τQMQ(P ∧ Q) = trQMQ 

(k) (k) 

B 1 ∧ PT B 2 

 

∧ Q 

(k) (k) 

B 1 ∧ PT |L B 2 

= P range( ∞k=1 eS| L (B (k) 

1 )eT | L (B (k) 

2 )). 

= 

= 

∞ 

k=1 

∞ 

k=1 

∞ 

k=1 

eS|L 

 

trQMQ eS|L 

 

τQMQ PS|L 

= μS|L,T |L (B) 

= 1. 

Thus, P ∧ Q = Q, and this shows that (c) holds. 

(k) (k) 

B 1 

eT |L B 2 

 

(k) 

B 1 

eT |L 

(k) 

B 1 ∧ PT |L 

(k) 

B 2 

(k) 

B 2


As mentioned in Remark 5.2, every open set U ⊆ C2 may be written as a union of countably 

many mutually disjoint sets from K. Thus, we have now proved existence of PS,T (U) for every 

such U, and for general B ∈ B(C2 ) we will define 

 

PS,T (B) := 

PS,T (U). (5.7) 

Then again, P := PS,T (B) satisfies that 

(a) P is S- and T -invariant. 

Moreover, we prove that with K = P(H), 

B⊆U,U⊆C 2 open 

(b) μS|K,T |K is concentrated on B, and 

(c) P is maximal with respect to the properties (a) and (b). 

These properties will entail that when B happens to be a union of countably many mutually 

disjoint sets from K, then (5.7) agrees with the previous definition of PS,T (B) (cf. (5.5)). 

Now, to see that (b) holds, note that μS|K,T |K is regular (cf. [6, Theorem 7.8]), and hence 

μS|K,T |K (B) = inf μS|K,T |K (U) | B ⊆ U, U ⊆ C2 open . (5.8) 

Let U be any open subset of C 2 containing B. Write U as a union of countably many mutually 

disjoint sets from K: 

Then, according to (5.6), 

μS|K,T |K 

(U) = 

U = 

∞ 

k=1 

∞ (k) 

B 

k=1 

1 

 

τP MP PS 

and using Proposition 2.8 and Lemma 3.3 we find that 

μS|K,T |K (U) = trP MP 

= τP MP 

∞ 

k=1 

∞ 

k=1 

= τP MP 

= τP MP (P ) 

= 1, 

eS|K 

PS|K 

 

× B(k) 

2 . 

(k) 

B 1 ∧ PT 

PS,T (U) ∧ P 

(k) 

B 2 ∧ P , 

(k) (k) 

B 1 

eT |K B 2 

 

(k) (k) 

B 1 ∧ PT |K B 2 

 

where PS,T (U) is given by (5.5). Hence by (5.8), μS|K,T |K is concentrated on B.


Finally, if Q ∈ M is any S- and T -invariant projection, and if μS|L,T |L is concentrated on B, 

where L = Q(H), then μS|L,T |L is concentrated on U for every open set U containing B. Hence, 

by the first part of the proof, Q PS,T (U) for every such U, and it follows from the definition 

of PS,T (B) that Q P . 

Concerning (5.4), note that if B = B1 × B2, where B1,B2 ∈ B(C), then, by the definitions 

of μS,T and PS,T (B), (5.4) holds. If B is a disjoint union of sets (B (k) ) ∞ k=1 = (B(k) 

1 × B(k) 

2 )∞ k=1 , 

where B (k) 

i ∈ B(C), k ∈ N, i = 1, 2, then 

μS,T (B) = 

∞ 

τ (k) 

PS,T B 1 

k=1 

Applying Proposition 2.8 we thus find that 

 

× B(k) 

2 = 

 

∞ 

μS,T (B) = τ supp 

k=1 

k=1 

eS 

∞ 

τ supp (k) (k) 

eS B 1 

eT B 2 . 

k=1 

(k) (k) 

B 1 

eT B 2 

 

 

∞ 

= τ supp (k) (k) 

eS B 1 

eT B 2 

 

= τ 

∞ 

k=1 

PS,T 

= τ PS,T (B) . 

B (k) 

1 

Finally, for general B ∈ B(C 2 ), since μS,T is regular, 

 

× B(k) 

2 

 

μS,T (B) = inf μS,T (U) | B ⊆ U ⊆ C 2 ,Uopen 

= inf τ PS,T (U) | B ⊆ U ⊆ C 2 ,Uopen 

 

 

 

= τ 

PS,T (U) 

B⊆U⊆C 2 ,Uopen 

= τ PS,T (B) . ✷ 

The proof given above may clearly be generalized to the case of n commuting operators 

T1,...,Tn ∈ M, so that Theorem 5.1 has a slightly more general version: 

Theorem 5.3. Let n ∈ N, letT1,...,Tn ∈ M be commuting operators, and let B ⊆ Cn be any 

Borel set. Then there is a maximal closed subspace, K = KT1,...,Tn (B), affiliated with M which 

is Ti-invariant for every i ∈{1,...,n}, and such that the Brown measure μT1|K,...,Tn|K is concentrated 

on B. Let PT1,...,Tn (B) ∈ M denote the projection onto KT1,...,Tn (B). Then more precisely: 

(i) if B = B1 ×···×Bn with Bi ∈ B(C), then 

PT1,...,Tn 

(B) = 

n 

PTi (Bi); (5.9) 

i=1


(ii) if B is a disjoint union of sets (B (k) ) ∞ k=1 

k ∈ N, i = 1,...,n, then 

(iii) and for general B ∈ B(C n ), 

Moreover, for every B ∈ B(C n ), 

PT1,...,Tn 

PT1,...,Tn 

(B) = 

(B) = 

6. An alternative characterization of μS,T 

= (B(k) 

1 ×···×B(k) n ) ∞ k=1 , where B(k) 

i ∈ B(C), 

∞ 

k=1 

 

B⊆U, U⊆C n open 

(k) 

PT1,...,Tn, B ; (5.10) 

PT1,...,Tn (U). (5.11) 

μT1,...,Tn (B) = τ PT1,...,Tn (B) . (5.12) 

In this final section we are going to give a characterization of the Brown measure of two 

commuting operators in M, which is different from the one we gave in Theorem 4.1. Recall 

from [4] that for T ∈ M, the Brown measure of T , μT , is the unique compactly supported Borel 

probability measure on C which satisfies the identity 

τ log |T − λ1| 

= log |z − λ| dμT (z) (6.1) 

for all λ ∈ C. 

We are going to prove that a similar property characterizes μS,T . 

C 

Theorem 6.1. Let S,T ∈ M be commuting operators. Then μS,T is the unique compactly supported 

Borel probability measure on C2 which satisfies the identity 

τ log |αS + βT − 1| 

= log |αz + βw − 1| dμS,T (z, w) (6.2) 

for all α, β ∈ C. 

C 2 

Remark 6.2. Let S,T ∈ M be as in Theorem 6.1. Note that if μS,T satisfies (6.2) for all α, β ∈ C, 

then for all α, β, λ ∈ C, 

τ log |αS + βT − λ1| 

= log |αz + βw − λ| dμS,T (z, w). (6.3) 

C 2 

This is clear for λ = 0, and for λ = 0, (6.3) follows from the fact that two subharmonic functions 

defined in C coincide iff they agree almost everywhere with respect to Lebesgue measure. It now 

follows from Brown’s characterization of μαS+βT that μαS+βT is the push-forward measure να,β 

of μS,T via the map (z, w) ↦→ αz + βw. On the other hand, if να,β = μαS+βT , then (6.2) holds.


Recall from [7] that the modified spectral radius of T ∈ M, r ′ (T ), is defined by 

r ′ (T ) := max |z| z ∈ supp(μT ) . (6.4) 

Also recall from [7, Corollary 2.6] that in fact 

r ′ 

(T ) = lim lim T 

p→∞ n→∞ 

n 1/n 

 

. (6.5) 

p/n 

Lemma 6.3. Let S,T ∈ M be commuting operators. Then the modified spectral radii, r ′ (S), 

r ′ (T ), r ′ (ST ) and r ′ (S + T), satisfy the inequalities 

and 

r ′ (ST ) r ′ (S) · r ′ (T ), (6.6) 

r ′ (S + T) r ′ (S) + r ′ (T ). (6.7) 

Proof. (6.6) follows from (6.5) and the generalized Hölder inequality (cf. [5]): for A,B ∈ M 

and for 0


Repeating this argument, we find that for arbitrary θ ∈[0, 2π[, 

i.e. 

Since θ was arbitrary, we conclude that 

and this proves (6.7). ✷ 

supp(μ e iθ (S+T) ) ⊆ z ∈ C Re z r ′ (S) + r ′ (T ) , 

supp(μS+T ) ⊆ z ∈ C Re e −iθ z r ′ (S) + r ′ (T ) . 

supp(μS+T ) ⊆ B 0,r ′ (S) + r ′ (T ) , 

Lemma 6.4. Let S,T ∈ M be commuting operators, and let α, β ∈ C. Then μαS,βT is the pushforward 

measure of μS,T via the map hα,β : C × C → C × C given by 

hα,β(z, w) = (αz, βw). 

Proof. Recall that μαS,βT is uniquely determined by the property that for all B1,B2 ∈ B(C), 

Now, it is easily seen that for α = 0 and β = 0, 

Hence, 

μαS,βT (B1 × B2) = τ PαS(B1) ∧ PβT (B2) . (6.8) 

PαS(B1) = PS 

 

1 

α B1 

 

 

1 

μαS,βT (B1 × B2) = τ PS 

α B1 

 

1 

∧ PT 

and PβT (B2) = PT 

β B2 

 

1 

β B2 

 

. 

 

1 

= μS,T 

α B1 × 1 

β B2 

 

−1 

= μS,T hα,β (B1 × B2) . (6.9) 

If for instance α = 0, then PαS(B1) = 0if0/∈ B1 and PαS(B1) = 1 if 0 ∈ B1. It then follows that 

(6.9) holds in this case as well. Similar arguments apply if β = 0. ✷ 

Proof of Theorem 6.1. As noted in Remark 6.2, it suffices to prove that for all α, β ∈ C, μαS+βT 

is the push-forward measure of μS,T via the map (z, w) ↦→ αz + βw. At first we will consider 

the case α = β = 1. Define a : C × C → C by 

We are going to prove that for all B ∈ B(C), 

a(z,w) = z + w (z,w∈ C). 

−1 

μS+T (B) = μS,T a (B) . (6.10)


It suffices to show that for every open set U ⊆ C, 

−1 

μS+T (U) μS,T a (U) . (6.11) 

Indeed, if this holds, then by regularity of μS+T and a(μS,T ), for every Borel set B ⊆ C, 

μS+T (B) = inf μS+T (U) 

B ⊆ U, U open 

inf a(μS,T )(U) 

B ⊆ U, U open 

= μS,T 

a −1 B . 

Since both measures are probability measures, and the above inequality holds for both B and B c , 

we must have identity. That is, (6.10) holds. 

Now, let U ⊆ C be any open set. Then V := a −1 (U) is open in C 2 and we may write V as a 

countable union of mutually disjoint “boxes,” 

where for z ∈ C and δ>0, 

V = 

∞ 

I(zn,δn) × I(wn,δn), 

n=1 

I(z,δ):= w ∈ C Re(z) − δ


r ′ S + T − (zn + wn)1 

′ 

r P(H) 

[S − zn1] 

′ 

+ r P(H) 

[T − wn1] 

 

P(H) 

r ′ [S − zn1] 

′ 

+ r PS(I (zn,δn))(H) 

[T − wn1] 

 

PT (I (wn,δn))(H) 

2 √ 2 δn, 

and it follows that μS+T |P(H) is concentrated on B(zn + wn, 2 √ 2δn) ⊆ U. Hence, P 

PS+T (U), and we are done. 

Now, if α, β ∈ C, then we conclude from the above and Lemma 6.4 that 

−1 −1 −1 

μαS+βT (B) = μαS,βT a (B) = μS,T hα,β a (B) = μS,T (a ◦ hα,β) −1 (B) , 

and since (a ◦ hα,β)(z, w) = αz + βw, this completes the proof of the identity (6.2). 

To prove uniqueness of μS,T , suppose that ν is a compactly supported Borel probability measure 

on C2 which satisfies the identity (6.2) for all α, β ∈ C. That is, for all α, β ∈ C, μαS+βT is 

the push-forward measure of ν via the map (z, w) ↦→ αz + βw. Then, to prove that ν = μS,T ,it 

suffices to prove that for all y = (y1,...,y4) ∈ R4 , 

 

e i(y,x) 

dμS,T (x) = e i(y,x) dν(x) (6.15) 

R 4 

(here we identify C with R 2 ). For x = (x1,...,x4) ∈ R 4 and y = (y1,...,y4) ∈ R 4 , note that 

R 4 

(y, x) = Re (y1 − iy2)(x1 + ix2) + (y3 − iy4)(x3 + ix4) , 

and hence with α = y1 − iy2 and β = y3 − iy4 we find that 

 

e i(y,x) 

dμS,T (x) = e iRe(αz+βw) 

dμS,T (z, w) = 


C 2 

 

= 

C 2 

R 4 

e iRe(αz+βw) 

dν(z,w) = 

R 4 

C 

e iRez dμαS+βT (z) 

e i(y,x) dν(x), 

Remark 6.5. In the proof above it was shown that for U ⊆ C an open set, we have the following 

inequality: 

−1 

PS+T (U) PS,T a (U) . (6.16) 

But it was also shown that the two projections above have the same trace: 

τ PS+T (U) −1 −1 

= μS+T (U) = μS,T a (U) = τ PS,T a (U) . 

Hence, the two projections in (6.16) are identical, and by Theorem 5.1(iii), for every Borel set 

B ⊆ C, we must have that 

−1 

PS+T (B) = PS,T a (B) . (6.17)


As in the previous section, one can easily generalize the proof given above to the case of an 

arbitrary finite set of commuting operators, {T1,...,Tn}. That is, we actually have the following 

alternative description of μT1,...,Tn . 

Theorem 6.6. Let n ∈ N, and let T1,...,Tn be mutually commuting operators in M. Then 

μT1,...,Tn is the unique compactly supported Borel probability measure on Cn which satisfies 

the identity 

τ log |α1T1 + ··· +αnTn − 1| 

= log |α1z1 + ··· +αnzn − 1| dμT1,...,Tn (z1,...,zn) (6.18) 

for all α1,...,αn ∈ C. 

Lemma 6.7. Define a : C 2 → C by 

C n 

a(z,w) = z + w, 

and let U ⊆ C be an open set. Then for every pair (S, T ) of commuting operators in M,wemay 

write V := a−1 (U) as a countable disjoint union of sets (I (zn,δn) × I(wn,δn)) ∞ n=1 , where for 

z ∈ C and δ>0, 

I(z,δ):= w ∈ C Re(z) − δ


Remark 6.8. Let S ∈ M be invertible, and let T1,...,Tn be mutually commuting operators 

in M. Then 

μ S −1 T1S,...,S −1 TnS 

= μT1,...,Tn . (6.20) 

Indeed, this follows from the characterization of μT1,...,Tn given in Theorem 6.6 and from the fact 

that for all T ∈ M, μS−1TS = μT (cf. [4]). 

Proposition 6.9. Let S,T ∈ M be commuting operators. Then μST is the push-forward measure 

of μS,T via the map m : (z, w) ↦→ zw. 

Proof. The proof is essentially the same as the one we gave above when considering the map 

a : (z, w) ↦→ z + w. Again it suffices to show that for every open set U ⊆ C, 

−1 

μST (U) μS,T m (U) , (6.21) 

and for such an open set U we write V := m −1 (U) as a countable union of mutually disjoint 

“boxes” as in (6.12), but this time we make sure that δn > 0 is so small that 

B znwn, √ 

2 δn T +|zn| ⊆ U. (6.22) 

As in the previous case, one only has to show that for every n ∈ N, 

 

PST (U) PS I(zn,δn) 

∧ PT I(wn,δn) . 

Fix n ∈ N and set P = PS(I (zn,δn)) ∧ PT (I (wn,δn)). Since 

r ′ [S − zn1] 

′ 

r P(H) 

[S − zn1] √ 

2 δn, 

PS(I (zn,δn))(H) 

and 

and since 

r ′ [T − wn1] 

′ 

r P(H) 

[T − wn1] √ 

2 δn, 

PT (I (wn,δn))(H) 

ST − znwn1 = (S − zn1)T + zn(T − wn1), 

we have (cf. Lemma 6.3) that 

r ′ [ST − znwn1] 

′ 

r P(H) 

(S − zn1)T 

+|zn|r P(H) 

′ [T − wn1] 

 

P(H) 

r ′ [S − zn1] 

T +|zn|r P(H) 

′ [T − wn1] 

 

P(H) 

√ 

2 δn T +|zn| . 

Thus, μST |P(H) is concentrated on B(znwn, √ 2δn(T +|zn|)) ⊆ U, and therefore P PST (U), 

as desired. ✷


Remark 6.10. As in the additive case, we infer from the proof given above that for every Borel 

set B ⊆ C we have: 

−1 

PST (B) = PS,T m (B) . (6.23) 

Proposition 6.11. Consider type II1 factors M1 and M2 with faithful tracial states τ1 and τ2, 

respectively. Let S ∈ M1 and T ∈ M2. Then 

and it follows that 

μS⊗1,1⊗T = μS ⊗ μT , (6.24) 

μS⊗1+1⊗T = μS ∗ μT , (6.25) 

μS⊗T = μS ⋆μT , (6.26) 

where ∗ (⋆, respectively) denotes additive (multiplicative, respectively) convolution. 

Proof. μS⊗1+1⊗T (μS⊗T , respectively) is the push-forward measure of μS⊗1,1⊗T via the map 

a : (z, w) ↦→ z + w (m : (z, w) ↦→ zw), respectively), and μS ∗ μT (μS ⋆μT , respectively) is the 

push-forward measure of μS ⊗μT via that same map. Thus, (6.25) and (6.26) follow from (6.24). 

To see that the latter holds, let B1,B2 ∈ B(C). It is easily seen that 

Hence, 

This proves (6.24). ✷ 

PS⊗1(B1) = PS(B1) ⊗ 1 and P1⊗T (B2) = 1 ⊗ PT (B2). 

μS⊗1,1⊗T (B1 × B2) = (τ1 ⊗ τ2) PS(B1) ⊗ 1 ∩ 1 ⊗ PT (B2) 

 

= τ1 PS(B1) 

τ2 PT (B2) 

= μS(B1)μT (B2) 

7. Polynomials in n commuting variables 

In this final section we will prove: 

= (μS ⊗ μT )(B1 × B2). 

Theorem 7.1. Let n ∈ N, and let q be a polynomial in n commuting variables, i.e. q ∈ 

C[z1,...,zn]. Then for every n-tuple (T1,...,Tn) of commuting operators in M, one has that 

μq(T1,...,Tn) = q(μT1,...,Tn ), (7.1) 

where q(μT1,...,Tn ) is the push-forward measure of μT1,...,Tn via q : Cn → C. 

The proof relies on the previous sections and a few technical lemmas.


Lemma 7.2. Given n ∈ N and commuting operators T1,...,Tn ∈ M. Let 1 i


Lemma 7.3. Let n ∈ N and let α ∈ C. Define an, m (α) 

n : C n+1 → C by 

an(z1,...,zn,zn+1) = (z1,...,zn + zn+1), (7.6) 

m (α) 

n (z1,...,zn,zn+1) = (z1,...,αznzn+1). (7.7) 

Then for any (n + 1)-tuple (T1,...,Tn+1) of commuting operators in M one has that 

and 

μT1,...,Tn−1,Tn+Tn+1 

μT1,...,Tn−1,αTnTn+1 

= an(μT1,...,Tn+1 ) (7.8) 

= m(α) n (μT1,...,Tn+1 ). (7.9) 

Proof. The proof is based on Lemma 7.2 and the fact that by (6.17) and (6.23), for any Borel set 

B ⊆ C we have 

−1 

PTn+Tn+1 (B) = PTn,Tn+1 1 (B) (7.10) 

and 

−1 (α) 

PαTnTn+1 (B) = PαTn,Tn+1 m (B) = PTn,Tn+1 m −1 

(B) . 

In order to prove (7.8), consider arbitrary Borel sets B1,...,Bn ⊆ C. We must show that 

μT1,...,Tn−1,Tn+Tn+1 (B1 ×···×Bn) = μT1,...,Tn+1 

a −1 

n (B1 ×···×Bn) , 

i.e. that 

μT1,...,Tn−1,Tn+Tn+1 (B1 

 

×···×Bn) = μT1,...,Tn+1 B1 ×···×Bn−1 × a −1 (Bn) . (7.11) 

But by Lemma 7.2 and by (7.10), 

PT1,...,Tn−1,Tn+Tn+1 (B1 ×···×Bn) = PT1 (B1) ∧···∧PTn−1 (Bn−1) ∧ PTn+Tn+1 (Bn) 

= PT1 (B1) ∧···∧PTn−1 (Bn−1) 

−1 

∧ PTn,Tn+1 a (Bn) 

 

B1 ×···×Bn−1 × a −1 (Bn) , 

= PT1,...,Tn+1 

and this proves (7.11). (7.9) follows in a similar way. ✷ 

Lemma 7.4. Let n ∈ N, and let σ ∈ Sn (the group of permutations of {1, 2,...,n}). Then for any 

n-tuple (T1,...,Tn) of commuting operators in M, 

μTσ(1),...,Tσ(n) 

= σ(μT1,...,Tn ), (7.12) 

where identify σ with the corresponding permutation of coordinates C n → C n . 

Proof. This follows easily from Theorem 4.1. ✷


Lemma 7.5. For n ∈ N and 1 i n define fi : C n → C n+1 by 

fi(z1,...,zn) = (z1,...,zn,zi). 

Then for n commuting operators T1,...,Tn ∈ M, one has that 

μT1,...,Tn,Ti 

Proof. Given Borel sets B1,...,Bn+1 ⊆ C we must show that 

Clearly, 

μT1,...,Tn,Ti (B1 ×···×Bn+1) = μT1,...,Tn 

= fi(μT1,...,Tn ). (7.13) 

f −1 

i (B1 ×···×Bn+1) . (7.14) 

f −1 

i (B1 ×···×Bn+1) = B1 ×···×(Bi ∩ Bn+1) ×···×Bn 

so that the right-hand side of (7.14) is 

τ PT1 (B1) ∧···∧PTi (Bi 

 

∩ Bn+1) ∧···∧PTn (Bn) 

= τ PT1 (B1) ∧···∧PTi (Bi) ∧ PTi (Bn+1) ∧···∧PTn (Bn) . 

But this is exactly the left-hand side of (7.14) and we are done. ✷ 

We will not give the proof of Theorem 7.1 in full generality but rather, by way of an example, 

illustrate how it goes. Consider for instance 3 commuting operators T1,T2,T3 ∈ M and the 

polynomial q ∈ C[z1,z2,z3] given by 

At first define φ1 : C 3 → C 5 by 

q(z1,z2,z3) = 1 + 2z 2 2 + z1z2z3. (7.15) 

φ1(z1,z2,z3) = (z2,z2,z1,z2,z3). 

By repeated use of Lemmas 7.4 and 7.5 we find that 

Next define φ2 : C 5 → C 2 by 

μT2,T2,T1,T2,T3 

= φ1(μT1,T2,T3 ). 

φ2(z1,...,z5) = (2z1z2,z3z4z5), 

and by repeated use of (7.9) and Lemma 7.4 conclude that 

With φ3 : C 2 → C given by 

μ 2T 2 2 ,T1T2T3 = (φ2 ◦ φ1)(μT1,T2,T3 ). 

φ3(z1,z2) = z1 + z2

we now have (cf. (7.8)) that 


μ(q−1)(T1,T2,T3) = (φ3 ◦ φ2 ◦ φ1)(μT1,T2,T3 ) = (q − 1)(μT1,T2,T3 ). 

It is now a simple matter to show that for all λ ∈ C, 

τ log q(T1,T2,T3) − λ1 

= log |z − λ| dq(μT1,T2,T3 )(z), 

and then by Brown’s characterization of μq(T1,T2,T3), μq(T1,T2,T3) = q(μT1,T2,T3 ), as desired. 


C 

Part of this work was carried out while I visited the UCLA Department of Mathematics. I 

want to thank the department, and especially the Operator Algebra Group, for their hospitality. 

I also thank my advisor, Uffe Haagerup, with whom I had enlightening discussions about this 

work. 

References 

[1] L. Aagaard, The non-microstates free entropy dimension of DT-operators, J. Funct. Anal. 213 (1) (2004) 176–205. 

[2] P. Ainsworth, Ubegrænsede operatorer affilieret med en endelig von Neumann algebra, Master thesis, University of 

Odense, 1985. 

[3] L. Breiman, Probability, Addison–Wesley, 1968. 

[4] L.G. Brown, Lidskii’s theorem in the type II case, in: H. Araki, E. Effros (Eds.), Geometric Methods in Operator 

Algebras, Kyoto, 1983, in: Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., 1986, pp. 1–35. 

[5] T. Fack, H. Kosaki, Generalized s-numbers of τ -measurable operators, Pacific J. Math. 123 (1986) 269–300. 

[6] G.B. Folland, Real Analysis, Modern Techniques and Their Applications, Wiley, 1984. 

[7] U. Haagerup, H. Schultz, Invariant subspaces of operators in a general II1-factor, preprint, 2005. 

[8] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. I, Academic Press, 1983. 

[9] K. Kuratowski, Topology, vol. I, second ed., Academic Press, London, 1966. 

[10] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974) 103–116.



Function spaces between BMO and critical 

Sobolev spaces ✩ 

Jean Van Schaftingen 

Département de Mathématique, Université Catholique de Louvain, 2 chemin du Cyclotron, 

1348 Louvain-la-Neuve, Belgium 

Received 29 November 2005; accepted 1 March 2006 



Abstract 

The function spaces Dk(Rn ) are introduced and studied. The definition of these spaces is based on a regularity 

property for the critical Sobolev spaces Ws,p (Rn ),wheresp = n, obtained by J. Bourgain, H. Brezis, 

New estimates for the Laplacian, the div–curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 

338 (7) (2004) 539–543 (see also J. Van Schaftingen, Estimates for L1-vector fields, C. R. Math. Acad. 

Sci. Paris 339 (3) (2004) 181–186). The spaces Dk(Rn ) contain all the critical Sobolev spaces. They are 

embedded in BMO(Rn ), but not in VMO(Rn ). Moreover, they have some extension and trace properties 

that BMO(Rn ) does not have. 


Keywords: Critical Sobolev spaces; BMO 


1.1. Integrals with divergence-free vector-fields 

When p

J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516 491 

p>nit is embedded in the space of Hölder continuous functions of exponent α, C 0,α (R n ), with 

α = 1 − n/p [1,5,19]. 

The case p = n is more delicate. When n>1, functions in W 1,n (R n ) do not need to be continuous 

or bounded, but have many properties in common with such functions. This is expressed 

for example by the embedding of W 1,n (R n ) in the spaces BMO(R n ) and VMO(R n ) of functions 

of bounded and vanishing mean oscillation [6]. These considerations are also valid for fractional 

Sobolev spaces W s,p (R n ), with sp = n. 

Another property of critical Sobolev space was recently obtained by Bourgain and Brezis [3, 

23]: for every vector field ϕ ∈ (L 1 ∩ C)(R n ; R n ) and u ∈ W s,p (R n ),ifdivϕ = 0 in the sense of 

distributions, then 

 

 

 

uϕ dx 

Cs,pϕL1 (Rn ) uWs,p (Rn ). (1.1) 

R n 

There is no such property for BMO(R n ) or for VMO(R n ) (see [2] and Remark 5.2). 

A natural question is the relationship between (1.1) and the embedding of W s,p (R n ) in the 

spaces BMO(R n ) and VMO(R n ). In order to answer it, we define, for n 1, the seminorm 

and the vector space 

uDn−1(R n ) = sup 

ϕ∈D(R n ;R n ) 

div ϕ=0 

ϕ L 1 (R n ) 1 

 

 

 

uϕ dx 

 

n 

Dn−1 R = u ∈ D ′ R n : uDn−1(Rn ) < ∞ . 

Here D(R N ; R N ) is the space of compactly supported smooth vector fields and D ′ (R n ) is the 

space of distributions [16]. The subscript n − 1 will be justified by further extensions. By the 

inequality (1.1), W s,p (R n ) is embedded in Dn−1(R n ). 

The question of the previous paragraph is answered as follows: VMO(R n ) is not embedded in 

Dn−1(R n ) (Proposition 5.1), and Dn−1(R n ) is embedded in BMO(R n ) (Theorem 5.3). Moreover, 

if u ∈ Dn−1(R n ) is continuous, and k 2, then u| R k BMO(R k ) CuDn−1(R n ) (Theorems 3.4 

and 5.3). This inequality remains open when k = 1. 

The proof of the embedding of Dn−1(R n ) in BMO(R n ) is based on the duality between 

BMO(R n ) and the Hardy space H 1 (R n ), and on a decomposition of every function in H 1 (R n ) as 

a sum of some components of divergence-free vector-fields, with a suitable control on the norms. 

The inequality (1.1) was preceded by a geometric counterpart [4]: for every closed rectifiable 

curve γ ∈ C 1 (S 1 ; R n ) and u ∈ (C ∩ W 1,n )(R n ), 

R n 

R n 

(1.2) 

 

 

 

u γ(t) 

 

˙γ(t)dt 

Cs,p˙γ L1 (S1 ) uWs,p (Rn ). (1.3)

492 J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516 

(See [22] for an elementary proof.) The right-hand side of (1.3) could also be used to define 

a seminorm on continuous functions. By the arguments of [3], based on a decomposition of 

divergence-free vector-fields in solenoids of Smirnov [18], one has in fact 


γ ∈C 1 (S 1 ;R n ) 

 

1 

 

˙γ u 

L1 (S1 ) 

γ(t) 

 

˙γ(t)dt 

. 

An open problem is whether restricting the curves on the right-hand side to be contained in 

k-dimensional planes, to triangles or to circles would yield an equivalent norm. The restriction 

to curves contained in k-dimensional planes is equivalent to requiring ϕ in (1.2) to have a range 

whose dimensions is at most k, see Section 6.4. 

If s 1, sp = n, and u ∈ W s,p (R n ), then u+ ∈ W s,p (R n ). This property also holds in 

BMO(R n ). We do not know whether it holds for Dn−1(R n ). The question whether, for a given 

ϕ : R → R one has ϕ(u) ∈ Dn−1(R n ) whenever u ∈ Dn−1(R n ) remains open when ϕ is not affine. 

1.2. Integrals with curl-free vector-fields 

When s = 1 and p = n = 2, the inequality (1.1) is in fact a dual statement of the Sobolev– 

Nirenberg embedding 

S 1 

g L n/(n−1) (R n ) CDg L 1 (R n ) . (1.4) 

When s = 1 and p = n>2, the estimate (1.1) is stronger than the embedding (1.4). If n = 3, 

(1.4) yields by duality that, for every ϕ ∈ D(R3 ; R3 ) and u ∈ W1,3 (R3 ), if curl ϕ = 0 in the sense 

of distributions, 

 

 

 

uϕ dx 

CϕL1 (R3 ) uWn (R3 ) . (1.5) 

R 3 

For u ∈ W s,p (R 3 ) with sp = 3, this inequality can be deduced from (1.1) recalling that, for every 

e ∈ R 3 

div(ϕ × e) = (curl ϕ) · e. (1.6) 

In R 3 , one can now investigate the relationship between (1.1), (1.5), and the embedding of 

W s,p (R n ) in the spaces BMO(R n ) and VMO(R n ). We define therefore the seminorm 


u D1(R 3 ) 

= sup 

ϕ∈D(R 3 ;R 3 ) 

curl ϕ=0 

ϕ L 1 (R 3 ) 1 

 

 

 

uϕ dx 

 

3 

D1 R = u ∈ D ′ R 3 : uD1(R3 ) < ∞ . 

R 3


While VMO(R3 ) ⊂ D1(R3 ), one has the following continuous embeddings: 

3 

D2 R 3 

⊂ D1 R ⊂ BMO R 3 . 

The first embedding is a consequence of (1.6), and the second of the duality between BMO(R3 ) 

and the Hardy space H1 (R3 ), and of a decomposition of every function in H1 (R3 ) as a sum of 

some components of curl-free vector-fields. 

If u ∈ D1(R2 ), its extension U(x,y) = u(x) to R3 is in D1(R3 ).ItwouldbeinD1(R3 ) 

if and only if U was bounded. On the other hand, if u ∈ D2(R3 ) is continuous, one has 

the trace inequality u| R2D1(R 2 ) CuD2(R3 ) . The problem whether the trace inequalities 

u| R2BMO(R2 ) CuD1(R3 ) and u|RBMO(R) CuD2(R3 ) hold is open. 

The seminorm ·D1(R3 ) can also be characterized geometrically: by the co-area formula, for 

every u ∈ C(R3 ), 

uD1(R3 1 

) = sup 

Ω H2 

 

 

(∂Ω) u(y)ν(y)dH 2 

 

(y) 

, 

where the supremum is taken over bounded domains Ω ⊂ R 3 with a smooth connected boundary, 

ν(y) is the unit exterior normal vector to the boundary at y ∈ ∂Ω, and H 2 is the two-dimensional 

Hausdorff measure. 

1.3. Integrals along differential forms 

In higher dimensions, the generalization of (1.1) corresponding to (1.5) in R3 is expressed 

with differential forms: if 1 k n − 1, then, for every compactly supported smooth 

k-differential form ϕ ∈ D(Rn ; ΛkRn ) and for every u ∈ Ws,p (Rn ) with p 1 and sp = n, if 

dϕ = 0, then 

 

 

 

 

 

 

uϕ dx 

Cs,pϕL1 (Rn ) uWs,p (Rn ). (1.7) 

R n 

The previous definitions of Dk(Rn ) are generalized as follows. For 1 k n − 1, we define 

the seminorm 

 

 

 

uϕ dx 

 


∂Ω 

uDk(R n ) = sup 

ϕ∈D(R n ;Λ k R n ) 

dϕ=0 

ϕ L 1 (R n ) 1 

R n 

n 

Dk R = u ∈ D ′ R n : uDk(Rn ) < ∞ . 

By (1.7), Ws,p (Rn ) ⊂ Dk(Rn ). These spaces Dk(Rn ) also contain other functions, such as 

log( k+1 i=1 x2 i ). 

The spaces Dk(Rn ) contain neither BMO(Rn ) nor VMO(Rn ). Our main result is that Dk(Rn ) 

is embedded in BMO(Rn ). We first show that Dk(Rn ) is embedded in D1(Rn ), then we prove


that D1(R n ) is embedded in BMO(R n ), in the same fashion as the embedding of D1(R 3 ) in 

BMO(R 3 ) outlined above. 

The other known properties of the spaces Dk(R n ) can be summarized as follows. The seminorm 

·Dk(R n ) is a norm modulo constants and the space Dk(R n ) modulo constants is a Banach 

space. The spaces Dk(R n ) are all different and are decreasing with respect to k. The spaces 

Dk(R n ) also have a trace property. If u ∈ DK(R N ) is a limit of continuous functions, then it 

has a well-defined trace in Dk(R n ) if N − K = n − k. On the other hand, the function spaces 

Dk(R n ) do not have better integrability properties then the exponential integrability of functions 

in BMO(R n ). 

1.4. Organization of the paper 

We define in Section 2 the spaces Dk(R n ) for 1 k n by duality on closed smooth forms, 

and characterize them by duality on exact forms (Proposition 2.6). The space Dn(R n ) is in fact 

L ∞ (R n )/R (Proposition 2.9). We characterize geometrically the seminorm for continuous functions 

in the cases k = 1 (Proposition 2.10) and k = n − 1 (Proposition 2.11). For 1

2.2. Definition 


The spaces Dk(R n ) are defined in terms of appropriate test function spaces. 

Definition 2.1. For 1 k n, define 

n k n 

D# R ; Λ R 

= ϕ ∈ D R n ; Λ k R n 

: dϕ = 0 and 

L 1 n k n 

# R ; Λ R 

= ϕ ∈ L 1 R n ; Λ k R n : dϕ = 0 and 

Remark 2.2. The restriction 1 k n, is justified by the fact that 

when k = 0ork>n. 

L 1 n k n 

# R ; Λ R n k n 

= D# R ; Λ R ={0} 

R n 

 

R n 

 

ϕdx= 0 , 

 

ϕdx= 0 . 

Remark 2.3. If 1 k n − 1, ϕ ∈ L 1 (R n ; Λ k R n ) and dϕ = 0, then 

R n ϕ = 0, while for every 

ϕ ∈ L 1 (R n ; Λ n R n ), dϕ = 0. Therefore, for a given 1 k 1, only one condition in the definition 

is essential. 

Definition 2.4. For 1 k n, and u ∈ D ′ (R n ),let 

and define 


ϕ∈D#(R n ;Λ k R n ) 

ϕ L 1 (R n ) 1 

 

 

 

uϕ dx 

, 

R n 

n 

Dk R = u ∈ D ′ R n : uDk(Rn ) < ∞ . 

The integral appearing in the definition of ·Dk(R n ) should be understood as a duality product 

between D ′ (R n ) and D(R n ; Λ k R), which takes its values in Λ k R n , while |·|is the standard 

Euclidean norm of this integral. The set Dk(R n ) is a vector space; the function ·Dk(R n ) is a 

seminorm on Dk(R n ), and vanishes for constant distributions. 

Remark 2.5. In fact, by Theorem 5.3, if uDk(R n ) = 0, then uBMO(R n ) = 0, whence u is 

constant. 

The seminorm ·Dk(R n ) can also be computed by considering exterior differentials of compactly 

supported smooth forms in place of closed compactly supported smooth forms. 

Proposition 2.6. Let 1 k n. For every u ∈ D ′ (Rn ), 

 

 

 

udψdx 

. 


ψ∈D(R n ;Λ k−1 R n ) 

dψ L 1 (R n ) 1 

R n


Proof. This follows from Theorems A.5 and A.8. ✷ 

Theorems A.5 and A.8 also allow to extend u by density to a linear operator from 

L 1 # (Rn ,Λ k R n ) to Λ k R n . 

Proposition 2.7. If u ∈ Dk(R n ), then 〈u, ϕ〉∈Λ k R n is well defined for every ϕ ∈ L 1 # (Rn ,Λ k R n ). 

We shall also consider the subspace generated by continuous functions. 

Definition 2.8. The space Vk(R n ) is the closure of the set of bounded continuous functions in 

Dk(R n ). 

2.3. Characterization of Dn(R n ) 

The space Dn(R n ) is well known; it is L ∞ (R n )/R. 

Proposition 2.9. The spaces Dn(R n ) and L ∞ (R n )/R are isometrically isomorphic. 

Proof. If u ∈ L∞ (Rn ), then for every ϕ ∈ D#(Rn ; ΛnRn ) and λ ∈ R, 

 

 

 

uϕ dx 

= 

 

 

 

 

 

(u − λ)ϕ dx 

u − λL∞ (Rn )ϕL1 (Rn ) . 

R 

R n 

Conversely, if u ∈ Dn(R n ), then 

 

ϕ ↦→ ℓ(ϕ) = 

R n 

uϕ dx 

is a linear continuous mapping from L1 # (Rn ,ΛnRn ) to ΛnRn ∼ = R. By the Hahn–Banach theorem 

there is an extension ¯ℓ to L1 (Rn ,ΛnRn ). This extension is represented as ¯ℓ(ϕ) = 

Rn ūϕ dx with 

ūL∞ (Rn ) uDn(Rn ). Since 

Rn ūϕ dx = 

Rn uϕ dx for every ϕ ∈ D#(Rn ,ΛnRn ), there is 

λ ∈ R such that u − λ =ū. ✷ 

2.4. Geometric characterization of V1(R n ) 

The definition of Dk(R n ), and hence that of Vk(R n ), rely on compactly supported closed 

smooth forms (Definition 2.4), or equivalently on compactly supported exact forms (Proposition 

2.6). Compactly supported smooth forms ensure that the definition makes sense for distributions. 

There is a more geometrical characterization for continuous functions, which extends by 

density to V1(R n ) and Vn−1(R n ). 

Proposition 2.10. For every u ∈ C(R n ), 

uD1(R n ) = sup 

Ω 

1 

Hn−1 

 

 

(∂Ω) 

∂Ω 

u(y)ν(y)dH n−1 

 

(y) 

, (2.1)


where the supremum is taken over bounded domains Ω with a smooth connected boundary, and 

ν(y) is the unit exterior normal vector to the boundary at y ∈ ∂Ω. 

Proof. Let Ω be a bounded domains with a smooth connected boundary. Let ρ ∈ D(B(0, 1)) be 

such that ρ 0 and 

R n ρdx= 1, let ρε(x) = ρ(x/ε)/ε n , and define 

Since ϕε1 1 and dϕε = 0, 

1 

ϕε(x) = 

Hn−1 

(∂Ω) 

R n 

∂Ω 

ρε(y − x)ν(y)dH n−1 (y). 

 

 

 

uϕε dx 

uD1(Rn ). 

Since u is continuous, ρε ∗ u → u as ε → 0 uniformly on every compact subset of Rn , and 

 

 

 

 

 

 

uϕε dx 

= 

1 

Hn−1 

 

 

(∂Ω) (ρε ∗ u)(y)ν(y) dH n−1 

 

(y) 

 

as ε → 0. Therefore 

R n 

∂Ω 

1 

→ 

Hn−1 

 

 

(∂Ω) u(y)ν(y)dH n−1 

 

(y) 

 

1 

Hn−1 

 

 

(∂Ω) 

∂Ω 

∂Ω 


 

(y) 

uD1(Rn ). 

Conversely, let A denote the right-hand side of (2.1). First note that for every bounded open 

set Ω with a smooth boundary that is not necessarily connected, 

 

 

 


 

(y) 

AHn−1 (∂Ω). 

∂Ω 

By Proposition 2.6, we need to evaluate, for ϕ ∈ D(Rn ,Λ0R), 

 

 

 

 

 

u∇ϕdx 

= 

 

 

 

 

 

 

ϕ∇udx 

. 

One has 

 

R n 

 

ϕ∇udx = 

∞ 

 

R n 

0 {x∈Rn : ϕ(x)>s} 

R n 

∇u(x)dxds − 

0 

 

−∞ {x∈Rn : ϕ(x)


For every s>0, the set {x ∈ Rn : ϕ(x) > s} is open and bounded. Moreover, by Sard’s lemma, 

for almost every s>0, for every y ∈ ϕ−1 ({s}), ∇ϕ(y) = 0. Hence ∂{x ∈ Rn : ϕ>s} is smooth 

and 

 

 

 

 

 

∇u(x) dx 

= 

 

 

 

 


 

(y) 

 

{x∈R n : ϕ>s} 

∂{x∈R n : ϕ(x)>s} 

AH n−1 ∂ x ∈ R n : ϕ(x) > s . 

A similar reasoning for s s + H n−1 ∂ x ∈ R n : ϕ(x) < −s ds 

 

= A |∇ϕ| dx. ✷ 

2.5. Geometric characterization of Vn−1(R n ) 

Proposition 2.11. If n 2, for every u ∈ C(R n ), 


γ ∈C 1 (S 1 ;R n ) 

 

1 

 

˙γ u 

L1 (S1 ) 

γ(t) 

 

˙γ(t)dt 

. 

The proof repeats the argument of Bourgain and Brezis for the equivalence between the inequality 

(1.7) and 

 

u γ(t) ˙γ(t)dtCs,p˙γ L1 (S1 ) uWs,p (Rn ), 

for every u ∈ W s,p (R n ) with sp = n [3]. 

Proof. First note that 

S 1 


f ∈D(R n ;R n ) 

div f =0 

f L 1 (R n ;R n ) 1 

S 1 

 

 

 

uf d x 

. 

Let ρ ∈ D(B(0, 1)) be such that ρ 0 and 

R n ρdx= 1, and let ρε(x) = ρ(x/ε)/ε n . Define 

R n 

 

1 

fε(x) = 

ρ 

˙γ L1 (S1 ) 

γ(t)−x ˙γ(t)dt. 

S 1


One has fεL1 (Rn ) 1 and div fε = 0, therefore 

 

 

 

ufε dx 

uDn−1(Rn ). 

R n 

The proof continues as for the first part of Proposition 2.10. 

The converse inequality comes from on a result of Smirnov [18], which states that for 

every R>0 and for every f ∈ D(B(0,R); Rn ) there exists (γ ℓ m )1m,ℓ in C1 (S1 ; B(0,R)) and 

(λℓ m )1m,ℓ in R such that for every m 1, 

 

λ ℓ 

 

m 

˙γ ℓ 

 

m L1 (S1 f ) L1 (Rn ) , 

and for every u ∈ C(B(0,R)) 

as m →∞. ✷ 

ℓ1 

∞ 

ℓ=1 

λ ℓ m 

2.6. Geometric characterization of Vk(R n ) 

 

S 1 

u γ ℓ m (t) ˙γ ℓ 

m dt → 

R n 

uf d x 

The characterization of the seminorm ·Dk(R n ) of Proposition 2.10, relies essentially on the 

fact that the seminorm could be evaluated by considering differential of scalar functions, while 

in Proposition 2.11 it relied on the decomposition result of Smirnov. Those facts do not hold 

anymore for 1


For every finite collection (xi)1ik in R n , the current ❏x0,...,xk❑ ∈ Dk(R n ) is defined by 

 

❏x0,...,xk❑,ϕ 

= 

Sk 

 

ϕ 

k 

x0 + λixi 

i=1 

 

 

,x1 − x0 ∧···∧xk − x0 dλ. 

Definition 2.13. A current T ∈ Dk(R n ) is a real polyhedral chain if there is (x i j )1im,1jk in 

R n and (μi)1im in R such that 

T = 

m 

i=1 

i 

μi x0 ,...,x i ② 

k . 

The set of k-dimensional real polyhedral chains is denoted by Pk(R n ). Every real polyhedral 

chain has a compact support and a finite mass. Hence, 〈T,u〉 is well defined when u : R n → Λ k R n 

is continuous. Moreover, if u : R n → R is continuous then 〈T,u〉∈(Λ k R n ) ′ ∼ = Λ k R n is naturally 

defined. 

Definition 2.14. If u : R n → R is continuous, let 

u ˜Vk(R n ) 

= sup 

P ∈Pn−k(R n ) 

∂P=0 

M(P )1 

〈P,u〉. 

The seminorm u ˜Vk(R n ) measures the oscillation of the function u through its integral on 

k-dimensional real polyhedral chains without boundary. 

Theorem 2.15. For every n 1 and 1 k n − 1, there exists c>0 such that for every u ∈ 

C(R n ), 

Proof. Given P in Pn−k(R n ),let 

u ˜Vk(R n ) uDk(R n ) cu ˜Vk(R n ) . 

ϕε(x) = P,ρε(·−x) , 

where ρε = ρ(·/ε)/ε n with ρ ∈ D(R n ), ρ 0 and 

R n ρdx= 1 and where ∗ denotes the Hodge 

duality between Λ k R n and Λ n−k R n . One checks that dϕε = 0, ϕε L 1 (R n ) M(P ) and 

 

R n 

Since ρε ∗ u → u uniformly as ε → 0, 

uϕε dx =∗〈P,ρε ∗ u〉. 

uVk(R n ) uDk(R n ).


The converse inequality is based on the deformation Theorem for currents [10,17]. It states 

that given T ∈ Dk(R n ) with M(T ) + M(∂T ) < ∞, for every ε>0, there exists P ∈ Pk(R n ), 

S ∈ Dk(R n ) and R ∈ Dk+1(R n ) such that 

with 

and 

T − P = ∂R + S, 

M(P ) cM(T ), M(∂P ) cM(∂T ), 

M(R) cεM(T ), M(S) cεM(∂T ), 

supp P ∪ supp R ⊂ x ∈ R n :dist(x, supp T)0 in Pk(R n ) with ∂Pε = 0 and supp Pε ⊂ U such that for every 

u ∈ C(R n ; R n ), 

and 

M(Pε) cM(T ), (2.2) 

〈Pε,u〉→〈T,u〉 (2.3) 

as ε → 0. 

Now given ϕ ∈ D#(Rn ; ΛkRn ), consider T ∈ Dn−k(Rn ) defined by 

 

〈T,v〉= ϕ ∧ vdx. 

R n 

Since T has compact support, ∂T = 0 and M(T ) ϕ L 1 (R n ) , there is a sequence (Pε)ε>0 in 

Pn−k(R n ) such that ∂Pε = 0, supp Pε ⊂ U, (2.2) and (2.3), where U is a fixed open bounded set 

such that supp T ⊂ U. ✷ 

3. Basic properties of Dk(R n ) 

3.1. Mutual injections 

The collection of spaces Dk(R n ) is a decreasing sequence of spaces. 

Theorem 3.1. Let k ℓ.Ifu ∈ Dℓ(R n ), then u ∈ Dk(R n ), and 

where C does not depend on u. 

uDk(R n ) CuDℓ(R n ),


Proof. Let ϕ ∈ D#(Rn ; ΛkRn ).Ifα ∈ Λℓ−kRn , then α ∧ ϕ ∈ D#(Rn ; ΛℓRn ). Therefore 

 

 

 

 

uα∧ ϕdx 

uDℓ(Rn )α ∧ ϕL1 (Rn ) uDℓ(Rn )|α|ϕL1 (Rn ) . 

R n 

Taking the supremum over α ∈ Λ ℓ−k R n with |α| 1 leads to the conclusion. ✷ 

3.2. Extension theory 

If n 0 are independent of u and U. 

R n 

R N−n 

ϕ(x,y)dy dx. 

cU Dk(R N ) uDk(R n ) CU Dk(R N ) , 

Proof. By induction, it is sufficient to consider the case N = n + 1. 

First let us estimate U Dk(R N ) . Consider Φ ∈ D#(R N ; Λ k R N ). It can be written as 

Φ = Φ0 + Φ1 ∧ ωN, 

where Φ0 ∈ D#(R N ; Λ k R n ) and Φ1 ∈ D#(R N ; Λ k−1 R n ). Define 

 

ϕ(x) = 

For m = 1, 2, 

and 

R 

 

Φ(x,t)dt, ϕ0(x) = 

dϕm(x) = 

 

R n 

n 

i=1 

R 

ωi ∧ ∂ϕm 

∂xi 

 

ϕm dx = 

 

Φ0(x, t) dt, ϕ1(x) = 

 

= 

 

Rn R 

R 

N 

i=1 

ωi ∧ ∂Φm 

∂xi 

Φm dt dx = 0. 

R 

dt = 0 

Φ1(x, t) dt.

Since ϕ = ϕ0 + ϕ1 ∧ ωN, one has 

 

UΦdz= 

and therefore, 

 

 

 

 

R N 


R N 

R n 

 

uϕ dx = 

R n 

 

uϕ0 dx + 

R n 

uϕ1 dx ∧ ωN 

 

 

UΦdz 

uDk(Rn )ϕ0L1 (Rn ) +uDk−1(Rn )ϕ1L1 (Rn ) |ωN |. 

When k>1, the conclusion comes from Theorem 3.1 and from the inequality 

ϕ0 L 1 (R n ) +ϕ1 L 1 (R n ) Φ0 L 1 (R n ) +Φ1 ∧ ωN L 1 (R n ) Φ0 + Φ1 ∧ ωN L 1 (R n ) . 

The last inequality comes from the fact that Φ0(x) and Φ1(x) ∧ ωN are orthogonal for every 

x ∈ RN . When k = 1, one has ϕ1 = 0, and the conclusion comes similarly. 

Conversely, let us now estimate uDk(Rn ) by Proposition 2.6. Let ψ ∈ D(Rn ; Λk−1Rn ). Consider 

a family (ηλ)λ>0 in D(R) such that ηλ 0, 

R ηλ dt = 1 and 

R |η′ λ | dt → 0asλ→∞, and let Ψλ(x, t) = ηλ(t)ψ(x). For every λ>0, 

 

R N 

Therefore, 

 

 

 

udψdx 

= 

 

 

 

 

R n 

 

UdΨλdz = 

R N 

R N 

Letting λ →∞yields the conclusion. ✷ 

 

U(dηλ∧ ψ + ηλdψ)dz = 0 + 

R n 

udψ dx. 

 

 

UdΨλdz UDk(R N 

) ψL1 (Rn ) +dψL1 (Rn ) η ′ λL1 

(R) . 

Remark 3.3. When kk(see Proposition 4.6). On the 

other hand, the extension of a function in Dn(R n ) lies in DN(R N ) by Proposition 2.9. In view of 

the trace theory of the next section, one could wonder whether when 1 kk. 

3.3. Trace theory 

The restriction of continuous functions from R N to R n can be extended to a continuous operator 

from VK(R n ) to Vk(R n ) when N − K = n − k. 

Theorem 3.4. Let n N, 1 K N and k = K − (N − n). Let U ∈ Vk(R N ) be continuous. 

Define for x ∈ R n , 

u(x) = U(x,0).


Then u ∈ Vk(R n ), and 

uDk(R n ) U DK(R N ) . 

Proof. By induction, we can assume N = n + 1. Let ϕ ∈ D#(Rn ; ΛkRn ),letρ∈D(R) such 

that ρ 0 and 

R ρdt = 1 and let ρε(t) = ρ(t/ε)/ε.LetΦε(x, t) = ρε(t)ψ(x) ∧ ωN . Since 

Φε ∈ D#(RN ; ΛKRN ) and u is continuous, 

 

 

 

 

 

 

uϕ dx 

= lim 

 

 

 

uΦε dt dx 

 

R n 

ε→0 

Rn R 

4. Examples of functions in Dk(R n ) 

4.1. Sobolev spaces 

UDK (RN ) lim 

ε→0 ΦεL1 (RN ) UDK (RN ) ϕL1 (Rn ) . ✷ 

The first class of functions in the space Dk(R n ) are functions in critical Sobolev spaces, which 

motivated the definition. 

Theorem 4.1. (Bourgain and Brezis [3]) If u ∈ W s,p (R n ), p>1 and sp = n, then for every 

1 k n − 1, u ∈ Dk(R n ), and 

uDk(R n ) Ck,s,puW s,p (R n ). 

The seminorm on the right-hand side is the Sobolev semi-norm. For 0


4.2. Locally Lipschitz functions in R n \{0} 

The space Dn−1(R n ) is larger than critical Sobolev spaces. There is a simple condition for 

locally Lipschitz functions in R n \{0} to be in Dn−1(R n ), which is satisfied e.g. by the function 

log |x|. 

Proposition 4.3. Let n 2 and u ∈ W 1,1 

loc (Rn \{0}).If|x|∇u ∈ L ∞ (R n ), then u ∈ Dn−1(R n ) and 

uDn−1(R n ) |x|∇u L ∞ (R n ) . 

Remark 4.4. In general, u/∈ Dn(R n ) as shows the function u(x) = log |x|. 

Proof. Let f ∈ D(Rn \{0}; Rn ) be such that div f = 0. Lemma 4.5 yields 

 

 

 

 

 

 

uf d x 

= 

 

 

 

 

 

 

x(∇u · f)dx 

 

∇u|x| L 

∞ (Rn f ) L1 (Rn ) . 

R n 

R n 

Therefore, for every ϕ ∈ D#(Rn \{0}; Rn ), 

 

 

 

uϕ dx 

∇u|x| 

L∞ (Rn ϕ ) L1 (Rn ) . 

Since {0} has vanishing n-capacity (Lemma A.2), 

R n 

 

n n−1 n n−1 

D# R \{0}; Λ R is dense in D# R ; Λ R 

(Theorem A.5). This concludes the proof. ✷ 

Lemma 4.5. Let u ∈ W 1,1 

loc (Rn \{0}) and f ∈ D(Rn \{0}; Rn ).Ifdiv f = 0, then 

 

 

uf d x =− x(f ·∇u) dx. 

Proof. By integration by parts, 

 

 

x(f ·∇u) dx =− 

R n 

R n 

R n 

R n 

 

x(div f)udx− 

The conclusion comes from the assumption div f = 0. ✷ 

4.3. Examples of functions in Dk(R n ) 

R n 

uf d x. 

Proposition 4.3 and Theorem 3.2 yield examples of functions in the spaces Dk(R n ) showing 

that these spaces are distinct.


Proposition 4.6. If 1 ℓ n, then 

if and only if 1 k


Corollary 4.9. (Bourgain and Brezis [3]) Let f ∈ L1 (R2 ; R2 ).Ifdiv f = 0 in the sense of distributions, 

then 

log 1 

|x| ∗ f ∈ L∞R 2 ; R 2 . 

Other interesting examples can be obtained in a similar way: 

Proposition 4.10. If 1 ℓ n and 0


and 

 

 

 

∂ 

 

2 

 

(Kn ∗ f) 

∂xi∂xj 

L 1 (R n ) 

Cf H 1 (R n ) . 

Finally, D(R n ) ∩ H 1 (R n ) is dense in H 1 (R n ). 

The space of functions with vanishing mean oscillations VMO(R n ; R n ) is the closed subspace 

of VMO(R n ) that is characterized by 

lim 

sup 

ε→0 Ln (B)ε 

1 

Ln (B) 2 

 

B 

 

B 

 

u(x) − u(y) dxdy = 0, 

where the supremum is taken over balls B ⊂ R n . The critical Sobolev spaces W s,p (R n ) with 

sp = n are embedded in VMO(R n ). 

Going back to the examples of the preceding section, for every ℓ 1, 

log 

ℓ 

i=1 

|xi| 2 

 

∈ BMO R n , 

but it does not belong to VMO(Rn ), while for 0


Proof. By Theorem 3.1, we can assume k = 1. The seminorm of u in BMO(R n ) will be estimated 

by duality with the Hardy space H 1 (R n ). 

Let f ∈ D(R n ) ∩ H 1 (R n ).Let 

ϕi = 

Note ϕj ∈ L 1 # (Rn ; Λ 1 R n ). Moreover, 

R n 

i=1 

R n 

n ∂2 (KN ∗ f)ωj . 

∂xi∂xj 

j=1 

 

 

 

fudx 

 

n 

 

 

 

∂ 2 f/∂x 2 i udx 

 

 

 

 

uD1(R n ) 

n 

 

 

 

 

i=1 

R n 

ϕi udx 

n 

ϕiL1 (Rn ) CuD1(Rn )f H1 (Rn ) . (5.1) 

i=1 

(Note that Proposition 2.7 about the well-definiteness of the duality product between D1(R n ) and 

L 1 # (Rn ; Λ 1 R n ) was used.) ✷ 

Remark 5.4. A similar argument shows that for 1


5.4. Integrability of functions in Dk(Rn ) 

If u ∈ BMO(Rn ), uBMO(Rn ) 1 and 

B(0,1) udx = 0, the John–Nirenberg theorem states 

that 

 

exp μ|u| dx c, (5.2) 

B(0,1) 

where μ>0 and c>0 can be chosen independently of u. Since the spaces Dk(Rn ) are embedded 

in BMO(Rn ), this might be improved on Dk(Rn ). 

On the other hand, if sp = n,0


The improvement of Dk(R n ) on BMO(R n ) should thus not be seen as an improvement of the 

integrand, but as an improvement on the domains of integration: by the trace Theorem 3.4, the 

embedding Theorem 5.3, and the John–Nirenberg inequality, functions in Vk(R n ) are exponentially 

integrable on (n − k + 1)-dimensional subspaces. 

6. Further problems 

6.1. Traces of V1(R n ) on VMO(R n−1 ) 

By Theorems 3.4 and 5.3, functions in Vk(R n ) have VMO traces on (n − k + 1)-dimensional 

spaces. The dimension n − k seems more natural: functions in Vn(R n ) are continuous, and hence 

have traces on 0-dimensional spaces, i.e. points. If there was such a trace inequality, one could 

define D0(R n ) = BMO(R n ). This notation would be consistent with the mutual injection Theorem 

3.1, the extension Theorem 3.2 and the examples of Proposition 4.6. It would then be nice 

to have a definition of Dk(R n ) which encompasses the case k = 0. The two-dimensional case 

would already solve the problem of traces of Vn−1(R n ) on lines. 

6.2. Geometric characterizations 

By Propositions 2.10 and 2.11, the spaces V1(R n ) and Vn−1(R n ) can be defined by oscillations 

respectively along boundaries of bounded domains and along closed curves. Further 

refinements would restrict the set of domains and of curves. The most striking result would be if 

the oscillation could be simply evaluated respectively on spheres and on circles. 

The spaces Vk(R n ) for 1


6.4. Similar spaces Ek(R n ) 

It would have been possible to define another family of spaces with properties similar to 

Dk(Rn ).For1k n,let 

n n 

R ; R = ϕ ∈ D R n ; R n :divϕ = 0 

D#,k 

and the dimension of the range of ϕ is at most k . 

(The set D#,k(Rn ; Rn ) is not a vector space when k


embedded respectively in bmoz( ¯Ω) (functions whose extension by 0 to R n is in BMO(R n )) and 

the second in the larger space bmor( ¯Ω) (restrictions of functions in BMO(R n ) to Ω) (see [8] for 

the definitions). 


The author thanks Haïm Brezis who suggested the problem, and who encouraged and discussed 

the progress of this work. He also thanks Thierry De Pauw for discussions, and acknowledges 

the hospitality of the Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie in 

Paris, at which this work was initiated. 

Appendix A. Density of compactly supported forms 

A.1. The closure of closed k-forms 

This appendix is devoted to the study of dense sets in the space L 1 (R n ; Λ k R n ) of summable 

closed forms. 

Definition A.1. Let 1 p0, the s-dimensional Hausdorff 

measure of Σ vanishes [9]. 

In a similar way, one can prove 

Lemma A.4. There exists a sequence (ζm)m in D(Rn ) such that 0 ζm 1, ζm → 1 almost 

everywhere and 

 

|∇ζm| n dx → 0 

as m →∞. 

R n


Theorem A.5. If Σ ⊂ R n is compact and has vanishing n-capacity, then d(D(R n \Σ; Λ k−1 R n )) 

is dense in L 1 # (Rn ; Λ k R n ). 

The proof makes use of a result of Bourgain and Brezis. 

Theorem A.6. (Bourgain and Brezis [3]) Let 1 k n − 1. For every ϕ ∈ L1 # (Rn ; ΛkRn ), there 

exists ψ ∈ Ln/(n−1) (Rn ; Λk−1Rn ) such that 

 

dψ = ϕ, 

δψ = 0. 

Here δ denotes the codifferential, i.e. the adjoint of d with respect to Hodge star. This result 

is based on inequality (1.7). When k = 1, the meaningless condition δψ = 0 is dropped and this 

is equivalent with the Nirenberg–Sobolev embedding. 

Proof of Theorem A.5. Since the exterior differential d commutes with translations, by classical 

smoothing arguments, (C ∞ ∩ L 1 # )(Rn ; Λ k R n ) is dense in L 1 # (Rn ; Λ k R n ). 

Let ϕ ∈ (C ∞ ∩ L 1 # )(Rn ; Λ k R n ) and let Σ ⊂ Ω ⊂ R n be open and bounded. Since Σ has 

vanishing capacity, there is a sequence (ηm)m1 in D(Ω) such that 0 ηm 1, ηm = 1ona 

neighborhood of Σ and ∇ηmL n (R n ) → 0asn →∞. Moreover, by Poincaré’s inequality, up to 

a subsequence, ηm → 0 almost everywhere. 

Consider now the sequence 

ψm = (1 − ηm)ζmψ, 

where ζm is given by Lemma A.4. By definition, ψm ∈ D(R n ; Λ k−1 R n ). We claim that dψm → ϕ 

in L 1 (R n ; Λ k R n ). 

In fact, 

By Hölder’s inequality, 

dψm =−ζm dηm ∧ ψ + (1 − ηm)dζm ∧ ψ + (1 − ηm)ζm ϕ. (A.1) 

−ζm dηm ∧ ψ L 1 (R n ) ζmL ∞ (R n )dηmL n (R n )ψ L n/(n−1) (R n ) . 

Since ∇ηmL n (R n ) → 0 and ψ L n/(n−1) (R n ) < ∞, the first term in (A.1) tends to zero. A similar 

reasoning holds for the second term, and the last term converges to ϕ as m →∞by Lebesgue’s 

dominated convergence theorem. ✷ 

Corollary A.7. The set D#(R n ; Λ k R n ) is dense in L 1 # (Rn ; Λ k R n ). 

A.2. The closure of exact n-forms 

Theorem A.6 fails when k = n, and therefore the proof Theorem A.5 fails in this case, but 

there is in fact a stronger result. 

Theorem A.8. The set d(D(R n ; Λ n−1 R n )) is dense in D#(R n ; Λ n R n ).


Remark A.9. The density is with respect to the usual topology on the space of test functions [16]. 

Proof of Theorem A.8. Let ϕ ∈ D#(R n ; Λ n R n ). Therefore ϕ = fω1 ∧ ··· ∧ ωn, with f ∈ 

D(R n ) and 

R n fdx= 0. Let (ρε)ε>0 be a sequence of mollifiers. Define gε ∈ D(R n ; R n ) by 

Next, note 

2 

gε(z) = 

f L1 (Rn ) 

2 

div gε(z) = 

f L1 (Rn ) 

2 

= 

f L1 (Rn ) 

 

R n ×R n 

 

R n ×R n 

 

R n ×R n 

 

(x − y) 

1 

0 

0 

1 

 

ρε z − tx − (1 − t)y f+(x)f−(y) dt dx dy. 

 

(x − y) ·∇ρε z − tx − (1 − t)y f+(x)f−(y) dt dx dy 

ρε(z − x) − ρε(z − y) f+(x)f−(y) dx dy 

= (ρε ∗ f )(z). (A.2) 

Therefore div gε → f in D(R n ) as ε → 0. Letting 

one concludes 

as ε → 0. ✷ 

ψε = 

n 

g i ε (−1)i+1ω1 ∧···∧ωi ∧···∧ωn, 

i=1 

dψε = div gε ω1 ∧···∧ωn → fω1 ∧···∧ωn = ϕ 

Remark A.10. The construction (A.2) is inspired from the construction of a non-optimal mass 

displacement plan in the Monge–Kantorovich mass displacement problem [11]. 

References 

[1] R.A. Adams, Sobolev Spaces, Pure Appl. Math., vol. 65, Academic Press, New York, 1975. 

[2] F. Bethuel, G. Orlandi, D. Smets, Approximations with vorticity bounds for the Ginzburg–Landau functional, Commun. 

Contemp. Math. 6 (5) (2004) 803–832. 

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A characteristic operator function 

for the class of n-hypercontractions ✩ 

Anders Olofsson 

Falugatan 22 1tr, SE-113 32 Stockholm, Sweden 

Received 5 December 2005; accepted 10 March 2006 




Abstract 

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates 

naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the 

unit disc. In the context of n-hypercontractions in the class C0· we introduce a counterpart to the so-called 

characteristic operator function for a contraction operator. This generalized characteristic operator function 

Wn,T is an operator-valued analytic function in the unit disc whose values are operators between two 

Hilbert spaces of defect type. Using an operator-valued function of the form Wn,T , we parametrize the wandering 

subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted 

Bergman space. The operator-valued analytic function Wn,T is shown to act as a contractive multiplier from 

the Hardy space into the associated standard weighted Bergman space. 


Keywords: Characteristic operator function; n-Hypercontraction; Wandering subspace; Standard weighted Bergman 

space; Reproducing kernel function 


Let us first describe a class of vector-valued standard weighted Bergman spaces that will play 

an important role in this paper. Let n 1 be an integer and let E be a general not necessarily 

✩ Research supported by the M.E.N.R.T. (France) and the G.S. Magnuson’s Fund of the Royal Swedish Academy of 

Sciences. 

E-mail address: ao@math.kth.se. 


doi:10.1016/j.jfa.2006.03.004

518 A. Olofsson / Journal of Functional Analysis 236 (2006) 517–545 

separable Hilbert space. We denote by An(E) the Hilbert space of all E-valued analytic functions 

f(z)= 

akz k , z∈D, (0.1) 

in the unit disc D with finite norm 

k0 

f 2 

An = 

k0 

ak 2 μn;k, 

where μn;k = 1/ k+n−1 k for k 0. Here the Taylor coefficients ak in (0.1) are elements in E. 

The weight sequence {μn;k}k0 is naturally identified as a sequence of moments of a certain 

radial measure dμn on the closed unit disc in the sense that 

 

μn;k = |z| 2k 

dμn(z) = 1 

 

k + n − 1 

, k0. k 

For n 2 the measure dμn is given by 

¯D 

dμn(z) = (n − 1) 1 −|z| 2 n−2 dA(z), z ∈ D, 

where dμ2(z) = dA(z) = dxdy/π, z = x + iy, is the usual planar Lebesgue area measure normalized 

so that the unit disc D is of unit area. The measure dμ1 is the normalized Lebesgue arc 

length measure on the unit circle T = ∂D. The norm of An(E) can also be expressed as 

f 2 An 

 

= lim 

r→1 

¯D 

 

f(rz) 2 dμn(z), f ∈ An(E). 

The shift operator Sn on the space An(E) is defined by 

(Snf )(z) = zf (z) = 

ak−1z k , z∈D, (0.2) 

k1 

for f ∈ An(E) given by (0.1). It is easy to see that the shift operator Sn is bounded on An(E) of 

norm equal to 1 (the weight sequence {μn;k}k0 is decreasing and the ratio μn;k+1/μn;k tends 

to 1 as k →∞). The adjoint operator S∗ n of Sn has the form 

∗ 

Snf (z) = μn;k+1 

ak+1z k , z∈D, (0.3) 

k0 

μn;k 

where the function f ∈ An(E) is given by (0.1). 

Let I be a shift invariant subspace of An(E). By this we mean that I is a closed subspace of 

An(E) which is invariant under the shift operator Sn in the sense that Sn(I) ⊂ I. The wandering 

subspace EI for I is the subspace 

EI = I ⊖ Sn(I)

A. Olofsson / Journal of Functional Analysis 236 (2006) 517–545 519 

of I. The subspace EI has the property that EI ⊥ S k n (EI) for k 1, which is often used as 

the defining property for a wandering subspace. The notion of a wandering subspace is often 

accredited to Halmos [12] and was used as an important concept in his description of the shift 

invariant subspaces of the Hardy space A1(E) using operator-valued inner functions. 

In the genuine Bergman space case n 2 it is known that the wandering subspace EI for a 

shift invariant subspace I of An(E) can have dimension equal to any positive integer or +∞ even 

in the case E = C of scalar-valued functions. This was first proved by Apostol et al. [5] using 

dual algebras, and later more explicit constructions have been found by Hedenmalm et al. [19] 

and others. 

In later developments the notion of a wandering subspace has proved to be a useful concept 

to study shift invariant subspaces in a Bergman space context. In the scalar case of invariant 

subspaces generated by zero sets Hedenmalm [13–15] has shown that functions in the wandering 

subspace also called Bergman inner functions can be used to divide out zeroes of functions in the 

subspace for n = 2, 3. For n = 1 we are in the Hardy space context, and for n 4 such a theorem 

fails (see [18]). 

A related question which has attracted much attention is to what extent the wandering subspace 

EI generates the whole invariant subspace I in the sense that 

I =[EI]= 

S k n (EI); (0.4) 

k0 

we use [F] to denote the smallest (closed) shift invariant subspace containing the set F. In our 

context of the Bergman spaces An(E) the approximation relation (0.4) is known to hold true for a 

general shift invariant subspace I of An(E) for the values n = 1, 2, 3, and is most probably false 

in general for n 4 (see [17,18]). The case n = 1 here is the Hardy space case mentioned earlier 

where a parametrization of the shift invariant subspaces is available. In the case of an unweighted 

Bergman space (n = 2) the approximation relation (0.4) for a general shift invariant subspace was 

first established by Aleman et al. [3] using function theoretic properties of the biharmonic Green 

function for the unit disc; see also [16, Section 3.6] and [20]. Later Shimorin [24] found a more 

general result which applies to a more general class of pure operators satisfying a certain operator 

inequality satisfied by the Bergman shift operator S2. The case n = 3 is due to Shimorin [25]. In 

the case n = 2 some summability results stronger than (0.4) are known to hold true (see [21]). 

Despite all the developments indicated above there are few explicit examples known of 

Bergman inner functions or, what is the same, wandering subspaces in the Bergman spaces. 

It is the purpose of the present paper to give a parametrization of the wandering subspace for a 

general shift invariant subspace in the context of the vector-valued standard weighted Bergman 

spaces An(E) described above. This parametrization is done in terms of certain operator theoretic 

quantities known as defect spaces, and some explicit formulas are obtained in the process. Let us 

now proceed to describe the content of the present paper. 

By a Hilbert space we mean a general not necessarily separable complex Hilbert space. We 

denote by L(H) the space of all bounded linear operators on a Hilbert space H. Letn 1bean 

integer. An operator T ∈ L(H) is called an n-hypercontraction if the operator inequality 

m 

k=0 

(−1) k 

 

m 

T 

k 

∗k T k 0 inL(H)


holds for every 1 m n. In this terminology a 1-hypercontraction is a contraction, and for 

n 2 the class of n-hypercontractions defines a more restricted class of operators. 

The class of n-hypercontractions was first introduced by Agler [1,2]. A principal result 

of [2] concerns the description of a general n-hypercontraction. An operator T ∈ L(H) is an 

n-hypercontraction if and only if it is unitarily equivalent to the restriction to an invariant subspace 

of an operator of the form S∗ n ⊕ U, where Sn is the shift operator on a Bergman space 

An(E) and U is an isometry. A special case of this result is the well-known fact that an operator 

T ∈ L(H) is a contraction if and only if it is part of an operator of the form S∗ 1 ⊕ U, where S1 

is the shift operator on a Hardy space A1(E) and U is an isometry, which is often accredited to 

Rota, de Branges, Rovnyak, Sz.-Nagy and Foias (see [26, Section I.10.1]). 

Let us recall that an operator T ∈ L(H) is said to belong to the class C0· if limk→∞ T k = 0 

in the strong operator topology meaning that limk→∞ T kx = 0inH for every x ∈ H (see [26, 

Section II.4]). For operators from the class C0· the isometric term U in the description in the 

previous paragraph vanishes and one has that an operator T ∈ L(H) is an n-hypercontraction 

such that limk→∞ T k = 0 in the strong operator topology if and only if it is a restriction of the 

adjoint shift operator S∗ n to an invariant subspace. 

We shall need some more notations related to an n-hypercontraction T ∈ L(H). We consider 

the defect operators 

Dm,T = 

m 

k=0 

(−1) k 

 

m 

T 

k 

∗k T k 

1/2 in L(H) 

for 1 m n, where the positive square root is used. The defect space Dm,T is defined as 

the closure in H of the range of the operator Dm,T , that is, Dm,T = Dm,T (H). Forn = 1 and 

T ∈ L(H) a contraction operator we write also 

DT = D1,T = (I − T ∗ T) 1/2 

in L(H) 

and DT = D1,T = DT (H) for the defect operator and the associated defect space. 

In recent work [22] we have revisited the operator model theory for the class of n- 

hypercontractions. It turns out that there is a canonical way to model an n-hypercontraction 

T ∈ L(H) as part of an operator of the form S∗ n ⊕ U, where U is an isometry. For x ∈ H we 

consider the Dn,T -valued analytic function Vnx defined by the formula 

(Vnx)(z) = Dn,T (I − zT ) −n x = 

 

k + n − 1 Dn,T T 

k 

k x z k , z∈D. (0.5) 

k0 

It turns out that if T ∈ L(H) is an n-hypercontraction such that limk→∞ T k = 0 in the strong 

operator topology, then the map Vn : x ↦→ Vnx given by (0.5) is an isometry 

Vn : H → An(Dn,T ) 

of H into An(Dn,T ) satisfying the intertwining relation 

VnT = S ∗ n Vn.


In this way an n-hypercontraction T ∈ L(H) in the class C0· is naturally modeled as part of the 

adjoint shift operator S ∗ n on the Bergman space An(Dn,T ). Full details of this construction can 

be found in [22, Sections 6 and 7]. 

We mention that construction of operator models of this type is a topic of current interest in 

multi-variable operator theory with recent contributions by Ambrozie et al. [4] and Arazy and 

Engliš [6]. Operator models of this type also form an integral part in recent work on constrained 

von Neumann inequalities by Badea and Cassier [8]. 

In this paper we shall consider in some more detail the subspace 

In,T = An(Dn,T ) ⊖ Vn(H) 

of An(Dn,T ). Since the range Vn(H) is invariant for S ∗ n , its orthogonal complement In,T is 

invariant for the shift operator Sn. In other words, the space In,T is a shift invariant subspace of 

An(Dn,T ). The wandering subspace En,T for In,T is the subspace 

En,T = In,T ⊖ Sn(In,T ) 

of In,T . To present our parametrization of the wandering subspace En,T for In,T we need some 

more notations. 

Let T ∈ L(H) be an n-hypercontraction. We denote by Hn the space H equipped with the 

equivalent norm 

x 2 n = 

n−1 

k=0 

(−1) k 

(see Lemma 3.1). It turns out that the operator 

TT ∗ 

 

n T n−1 

k 

x2 2 

=x + Dk,T x 

k + 1 

2 , x ∈ H (0.6) 

 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

 

k=1 

in L(H) 

is self-adjoint in L(Hn) and has its spectrum contained in the closed unit interval [0, 1] (see 

Lemma 3.3). We denote by Qn,T the operator 

Qn,T = 

 

I − TT ∗ 

 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

1/2 in L(H), 

where the positive square root is computed in L(Hn).ByD∗ n,T we denote the closure in H of the 

range of this operator Qn,T , and we equip this space D∗ n,T with the norm ·n defined by (0.6). 

It turns out that we have the equality 

T ∗ 

 

n−1 

(−1) k 

k=0 

in L(H) (see Lemma 3.4). 

 

n 

T 

k + 1 

∗k T k 

 

Qn,T = Dn,T T ∗ 

 

n−1 

k=0 

(−1) k 

 

n 

T 

k + 1 

∗k T k 

 

(0.7)


In the case n = 1 of a contraction operator T ∈ L(H) the notions in the previous paragraph 

specialize to well-known objects. The norm ·1 defined by (0.6) coincides with the usual norm 

of H. The operator Q1,T is the defect operator for the adjoint operator T ∗ , that is, Q1,T = DT ∗, 

and the space D ∗ 1,T is the defect space D∗ 1,T = DT ∗ for T ∗ . The equality (0.7) reduces to the 

well-known formula T ∗ DT ∗ = DT T ∗ for defect operators. 

For an n-hypercontraction T ∈ L(H) we consider the operator-valued analytic function Wn,T 

in the unit disc D defined by the formula 

Wn,T (z) = 

z ∈ D. 

 

−T ∗ 

 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

 

n 

+ zDn,T (I − zT ) 

k=1 

−k 

 

D 

Qn,T , 

∗ 

n,T 

Notice that by (0.7) the values Wn,T (z) attained by this function Wn,T are operators in 

L(D∗ n,T , Dn,T ), that is, bounded linear operators from D∗ n,T into Dn,T . 

We remark that in the case n = 1 of a contraction operator T ∈ L(H) we get the so-called 

characteristic operator function 

WT (z) = W1,T (z) = −T ∗ + zDT (I − zT ) −1 DT ∗ 

 

DT , z∈D, ∗ 

whose values are operators in L(DT ∗, DT ) which has been studied by Sz.-Nagy and Foias 

(see [26, Chapter VI]). 

We have the following description of the wandering subspace En,T . A function f in An(Dn,T ) 

belongs to the wandering subspace En,T for In,T if and only if it has the form 

f(z)= Wn,T (z)x, z ∈ D, (0.8) 

for some element x ∈ D∗ n,T . Furthermore, we have the norm equality 

f 2 An =x2 n , x ∈ D∗ n,T , 

when f is given by (0.8). We recall that the norm ·n is defined by (0.6). This parametrization 

of the wandering subspace En,T for In,T is the result of Theorem 3.3 in this paper. The proof of 

Theorem 3.3 proceeds in several steps and takes up Sections 2 and 3 in the paper. 

It turns out that the operator-valued analytic function Wn,T has a certain multiplier property 

in that it acts as a contractive multiplier from the Hardy space A1(D∗ n,T ) into the Bergman space 

An(Dn,T ) (see Theorem 4.1). This contractive multiplier property leads in turn to an estimate 

Wn,T (z)Wn,T (z) ∗ 

1 

(1 −|z| 2 ) n−1 IDn,T in L(Dn,T ), z ∈ D (0.9) 

(see Theorem 4.2). 

In the case n = 1 of a contraction operator T ∈ L(H) in the class C0· it is known that the 

characteristic operator function WT is an isometric multiplier from the Hardy space A1(DT ∗) into 

the Hardy space A1(DT ) with range equal to I1,T (see Corollary 4.1). Here the inequality (0.9) 

says that the characteristic operator function WT attains contractive values (see Corollary 4.2).


We mention also that the contractive multiplier property of Wn,T and the inequality (0.9) 

generalize to a vector-valued context known properties of Bergman inner functions going back 

to Hedenmalm [13–15] for n = 2, 3. 

As we have indicated above the previous considerations apply also to general shift invariant 

subspaces in the Bergman spaces An(E). LetI be a shift invariant subspace of An(E). Nowthe 

orthogonal complement 

H = An(E) ⊖ I 

of I is invariant for S ∗ n and we set T = S∗ n |H. The operator T ∈ L(H) is an n-hypercontraction in 

the class C0·. As above we can model this operator T by means of the map Vn given by (0.5). Furthermore, 

by a uniqueness property of this representation, there exists an isometry ˆVn : Dn,T → E 

such that every function f ∈ H admits the representation 

f(z)= ˆVnDn,T (I − zT ) −n f, z ∈ D. (0.10) 

The isometry ˆVn : Dn,T → E is uniquely determined by (0.10) and is given by 

ˆVn : Dn,T f ↦→ f(0) for f ∈ H. 

Full details of this construction can be found in [22, Sections 6 and 7]. 

We write Ê = ˆVn(Dn,T ) ⊂ E. Themap ˆVn naturally extends to an isometry 

ˆVn : An(Dn,T ) → An(E) 

of An(Dn,T ) into An(E) with range equal to An(Ê) by setting 

 

( ˆVnf )(z) = ˆVn f(z) , z∈D, for f ∈ An(Dn,T ). 

The shift invariant subspace I now decomposes as an orthogonal sum 

I = An(E ⊖ Ê) ⊕ ˆVn(In,T ) 

(see Theorem 5.1), and we can identify the wandering subspace EI for I as the orthogonal sum 

EI = (E ⊖ Ê) ⊕ ˆVn(En,T ). 

By our previous description of the wandering subspace En,T for In,T we have that a function f 

in An(E) belongs to the wandering subspace EI for I if and only if it has the form 

f(z)= a0 + ˆVnWn,T (z)g, z ∈ D, (0.11) 

for some elements a0 ∈ E ⊖ Ê and g ∈ D ∗ n,T . Furthermore, we have the norm equality f 2 An = 

a0 2 +g 2 n for f ∈ An(E) of the form (0.11) (see Theorem 5.2). 

As a byproduct of our considerations we obtain in an explicit form a parametrization of the 

shift invariant subspaces of the Hardy space A1(E) in case of a general not necessarily separable


Hilbert space E (see Corollaries 4.1 and 5.1). We discuss also some relations between the index 

of a shift invariant subspace and the defect indexes of the adjoint shift restricted to its orthogonal 

complement (see Corollary 5.2 and Proposition 5.1). 

We wish to mention that a source of inspiration for the work presented in this article has been 

the survey paper [9] by Ball and Cohen. 

1. Preliminaries 

Let us first recall some constructions developed in the context of so-called wandering subspace 

theorems in the papers [23,24]. The reason to include this discussion here is that it provides a 

motivation for some of the arguments we shall use in later sections. 

Let T ∈ L(H) be an injective operator. It is easy to see that then the following statements are 

equivalent: 

• The operator T has closed range T(H). 

• The operator T is bounded from below in the sense that Tx 2 cx 2 for x ∈ H and some 

positive constant c. 

• The operator T is left-invertible. 

Let now T ∈ L(H) be an operator satisfying any of these conditions. The wandering subspace 

for the operator T ∈ L(H) is the subspace 

E = H ⊖ T(H) = ker T ∗ 

of H. The operator L = (T ∗ T) −1 T ∗ in L(H) is the left-inverse of T with kernel E: 

The operator 

LT = I in L(H) and ker L = ker T ∗ = E. 

P = I − TL in L(H) 

is the orthogonal projection of H onto E. Indeed, the operator TL= T(T ∗ T) −1 T ∗ is self-adjoint, 

idempotent and has range equal to T(H). 

We shall also have use of the operator 

T ′ = L ∗ = T(T ∗ T) −1 

in L(H). 

The operator T ′ turns out to have some properties dual to those of T (see [21,24]). We notice 

that 

(T ′ ) ∗ T ′ = (T ∗ T) −1 T ∗ T(T ∗ T) −1 = (T ∗ T) −1 

in L(H). (1.1) 

Let us now specialize to the shift operator Sn on the Bergman space An(E). The formulas 

(0.2) and (0.3) make evident that the operator S∗ nSn on An(E) acts as 

 

∗ μn;k+1 

Sn Sn (f )(z) = akz 

μn;k 

k0 

k , z∈D,


where f ∈ An(E) is given by (0.1). We notice that 

Snf 2 An = S ∗ 

nSnf,f = An 

k0 

ak 2 μn;k+1, 

where f ∈ An(E) is given by (0.1). It is straightforward to verify that the quotient μn;k+1/μn;k is 

increasing in k 0. As a result we have that the operator Sn is bounded from below with constant 

c = 1/n. 

A computation shows that the operator Ln = (S ∗ n Sn) −1 S ∗ n on An(E) acts as 

(Lnf )(z) = 

f(z)− f(0) 

z 

= 

ak+1z k , z∈D, where f ∈ An(E) is given by (0.1). The operator S ′ n = L∗n = Sn(S∗ nSn) −1 is the weighted shift 

operator on An(E) acting as 

′ 

S nf (z) = μn;k−1 

ak−1z k , z∈D, (1.2) 

k1 

μn;k 

where f ∈ An(E) is given by (0.1). We notice that 

 

S ′ nf 2 

= An 

k1 

μ 2 n;k−1 

μn;k 

k0 

ak−1 2 = 

μ 2 n;k 

μn;k+1 

k0 

ak 2 , (1.3) 

where f ∈ An(E) is given by (0.1). 

Sums involving binomial coefficients will appear in our calculations. For the sake of easy 

reference we record the following lemma. 

Lemma 1.1. Let μn;k = 1/ k+n−1 k for n 1 and k 0. Then 

min(n−1,k) 

(−1) j 

j=0 

Proof. A computation shows that 

n−1 

(−1) k 

k=0 

 

n 

 

1 

j + 1 μn;k−j 

 

n 

z 

k + 1 

k = 

= 1 

. 

μn;k+1 

1 − (1 − z)n 

. 

z 

The sum in the lemma equals the kth Taylor coefficient of the function 

1 − (1 − z) n 

z 

 

 

1 1 1 

= − 1 

(1 − z) n z (1 − z) n 

This yields the conclusion of the lemma. ✷ 

= 

1 

μn;k+1 

k0 

z k , z∈ D.


We shall need the following norm equality. 

Proposition 1.1. Let f ∈ An(E) be given by (0.1). Then 

n−1 

(−1) k 

k=0 

Proof. By(0.3)wehavethat 

 

n S 

k + 1 

∗k 

n f 2 

= An 

k0 

μ 2 n;k 

μn;k+1 

 

S ∗j 

n f 2 μ 

= An 

k0 

2 n;k+j 

ak+j 

μn;k 

2 

ak 2 . 

for j 0. Summing these equalities we have by a change of order of summation that 

n−1 

(−1) j 

j=0 

 

n S∗j n f 

j + 1 

2 

= An 

k0 

= 

 

min(n−1,k) 

(−1) j 

μ 2 n;k 

j=0 

μn;k+1 

k0 

where the last equality follows by Lemma 1.1. ✷ 

ak 2 , 

 

n 1 

ak 

j + 1 μn;k−j 

2 μ 2 n;k 

We remark that the sums in Proposition 1.1 equal S ′ nf 2 by (1.3). By a polarization argu- 

An 

ment we conclude that 

∗ −1 ′ ∗S ′ 

Sn Sn = S n n = 

where the first equality follows by (1.1). 

n−1 

(−1) k 

k=0 

2. A first description of the wandering subspace 

 

n 

S 

k + 1 

k nS∗k n in L An(E) , (1.4) 

We use the same basic notations as in the introduction. The operator Vn in L(H,An(Dn,T )) 

is defined by (0.5), 

In,T = An(Dn,T ) ⊖ Vn(H) and En,T = In,T ⊖ Sn(In,T ). 

In this section we shall give a first description of the wandering subspace En,T for In,T in Theorem 

2.1. First we need a few lemmas. 

Lemma 2.1. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Then the operator 

Pn = I − VnV ∗ n 

is the orthogonal projection of An(Dn,T ) onto In,T . 

in L An(Dn,T )


Proof. We consider the operator Qn = VnV ∗ n . Recall that Vn : H → An(Dn,T ) is an isometry 

(see [22, Section 7]). It is straightforward to see that the operator Qn is self-adjoint, idempotent 

and has range equal to Vn(H). Accordingly the operator Qn is the orthogonal projection of 

An(Dn,T ) onto Vn(H), and Pn = I − Qn is the orthogonal projection of An(Dn,T ) onto In,T = 

An(Dn,T ) ⊖ Vn(H). ✷ 

We next compute the operator V ∗ n . 

Lemma 2.2. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Then the adjoint operator 

V ∗ n : An(Dn,T ) → H acts as 

V ∗ n 

 

f = T ∗k Dn,T ak weakly in H, (2.1) 

k0 

where f ∈ An(Dn,T ) is given by (0.1). 

Proof. For x ∈ H we have that 

∗ 

Vn f,x 

 

=〈f,Vnx〉An = ak, 1 

Dn,T T 

μn;k 

k x 

μn;k 

k0 

k0 

= 

ak,Dn,T T k x 

N 

= lim 

This gives the conclusion of the lemma. ✷ 

N→∞ 

k=0 

T ∗k Dn,T ak,x 

We remark that the sum in (2.1) converges in the weak topology in H. 

We shall next compute the operator V ∗ n (S∗ n Sn) −1 Vn = V ∗ n (S′ n )∗ S ′ n Vn. 

Lemma 2.3. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Then 

V ∗ ∗ −1Vn 

n Sn Sn = V ∗ ′ ∗S ′ 

n S n nVn = 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

 

. 

in L(H). 

Proof. Recall that the operator Vn in L(H,An(Dn,T )) is an isometry such that VnT = S ∗ n Vn 

(see [22, Sections 6 and 7]). By formula (1.4) we have that 

V ∗ ∗ −1Vn 

n Sn Sn = V ∗ ′ ∗S ′ 

n S n nVn = V ∗ n 

 

n−1 

(−1) k 

k=0 

 

n 

S 

k + 1 

k nS∗k 

n Vn in L(H). 

The intertwining relation VnT = S∗ nVn gives that VnT k = S∗k n Vn for k 0, and taking adjoints 

we see that also T ∗kV ∗ n = V ∗ n Sk n for k 0. Using these intertwining formulas we now have that


V ∗ ∗ 

n Sn Sn 

 

−1Vn 

= 

n−1 

(−1) k 

k=0 

n−1 

= 

k=0 

(−1) k 

 

n 

T 

k + 1 

∗k V ∗ k 

n VnT 

 

n 

T 

k + 1 

∗k T k 

in L(H), 

where the last equality follows by V ∗ n Vn = I in L(H). This completes the proof of the 

lemma. ✷ 

We can now give a first description of the wandering subspace En,T . 

Theorem 2.1. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Then a function f in 

An(Dn,T ) belongs to the wandering subspace En,T for In,T if and only if it has the form 

f = a0 + S ′ nVnx ∗ −1Vnx 

= a0 + Sn Sn Sn 

for some elements a0 ∈ Dn,T and x ∈ H such that 

Proof. Notice first that 

and similarly that 

Here 

Dn,T a0 + T ∗ 

 

n−1 

(−1) k 

k=0 

In,T = An(Dn,T ) ⊖ Vn(H) = ker V ∗ n , 

En,T = In,T ⊖ Sn(In,T ) = ker(Sn|In,T )∗ . 

(Sn|In,T )∗ = PnS ∗ n = I − VnV ∗ ∗ 

n Sn 

n 

T 

k + 1 

∗k T k 

 

x = 0. (2.2) 

by Lemma 2.1. An element f in An(Dn,T ) thus belongs to the wandering subspace En,T for In,T 

if and only if V ∗ n f = 0 and (I − VnV ∗ n )S∗ nf = 0. 

We consider first the equation (I − VnV ∗ n )S∗ nf = 0. This equation can be rewritten as S∗ nf = 

VnV ∗ n S∗ n f . We apply the operator (S∗ n Sn) −1 to obtain that 

Lnf = S ∗ nSn −1S∗ nf = S ∗ nSn −1VnV ∗ n S∗ nf = S ∗ nSn −1Vnx, where x = V ∗ n S∗ nf ∈ H. We now have that 

∗ −1Vnx 

f = a0 + SnLnf = a0 + Sn Sn Sn = a0 + S ′ nVnx, (2.3) 

where a0 ∈ Dn,T and x ∈ H. Conversely, if f in An(Dn,T ) is of the form (2.3), then 

 

I − VnV ∗ ∗ 

n Snf = I − VnV ∗ n Vnx = 0,


since Vn is an isometry. We have thus shown that a function f in An(Dn,T ) satisfies the equation 

(I − VnV ∗ n )S∗ nf = 0 if and only if it has the form (2.3) for some elements a0 ∈ Dn,T and x ∈ H. 

We shall now compute V ∗ n f when f ∈ An(Dn,T ) is of the form (2.3) for some elements a0 ∈ 

Dn,T and x ∈ H. Notice that the intertwining relation VnT = S∗ nVn gives that V ∗ n Sn = T ∗V ∗ n .By 

Lemma 2.2 we now have that 

V ∗ n f = Dn,T a0 + V ∗ n Sn 

∗ −1Vnx 

Sn Sn = Dn,T a0 + T ∗ V ∗ ∗ −1Vnx. 

n Sn Sn 

Recall that the operator V ∗ n (S∗ n Sn) −1 Vn was computed in Lemma 2.3. We conclude that 

V ∗ n f = Dn,T a0 + T ∗ 

 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

 

x, 

where f ∈ An(Dn,T ), a0 ∈ Dn,T and x ∈ H are related as in (2.3). This gives the conclusion of 

the theorem. ✷ 

We shall next compute the norm of a function of the form f = a0 + S ′ n Vnx. 

Theorem 2.2. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Let f in An(Dn,T ) be of 

the form 

for some elements a0 ∈ Dn,T and x ∈ H. Then 

Proof. By Lemma 2.3 we have that 

f = a0 + S ′ n Vnx 

f 2 An =a0 2 n−1 

+ 

k=0 

(−1) k 

f 2 An =a0 2 + ′ 

S nVnx 2 An =a0 2 n−1 

+ 

This completes the proof of the theorem. ✷ 

3. Parametrization of the wandering subspace En,T 

 

n T 

k + 1 

k x 2 . 

k=0 

(−1) k 

 

n T 

k 

x2 . 

k + 1 

In this section we shall solve Eq. (2.2) and give a more refined description of the wandering 

subspace En,T for In,T using the operator-valued analytic function Wn,T . 

We shall need some constructions involving the use of defect operators of contractions between 

Hilbert spaces. Let A ∈ L(H, K) be a contraction operator mapping a Hilbert space H 

into a Hilbert space K. Associated to this operator A we have the defect operator DA defined by 

DA = (I − A ∗ A) 1/2 

in L(H),


where the positive square root is used. Notice that 

x 2 =Ax 2 +DAx 2 , x ∈ H, (3.1) 

and that this equality (3.1) can be restated saying that I = A∗A + D2 A in L(H). The defect space 

DA for A is defined as the closure in H of the range of the operator DA, that is, DA = DA(H). 

In the same way the adjoint operator A∗ ∈ L(K, H) has an associated defect operator 

DA ∗ = (I − AA∗ ) 1/2 

in L(K), 

and a defect space DA∗ = DA∗(K) contained in K. These operators satisfy the equalities 

DA∗A = ADA in L(H, K) and DAA ∗ = A ∗ DA∗ in L(K, H). (3.2) 

The verification of the equalities (3.2) uses the functional calculus for self-adjoint operators in 

Hilbert space (see [10, Section XXVII.1] for details). 

Using the equalities (3.1) and (3.2) in the previous paragraph one verifies that the block oper- 

ator matrix 

 

A DA 

θA = 

∗ 

DA −A∗ 

: H ⊕ DA∗ → K ⊕ DA 

is a unitary operator. The construction of this unitary operator θA goes back to Halmos [11]. 

Let us now return to an n-hypercontraction T ∈ L(H). We shall need the following lemma. 

Lemma 3.1. Let T ∈ L(H) be an n-hypercontraction. Then 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k = I + 

Proof. For 1 m n we denote by Σm the operator 

m−1 

Σm = 

k=0 

(−1) k 

Clearly Σ1 = I .Form 2 we have that 

Σm − Σm−1 = (−1) m−1 T ∗(m−1) T m−1 m−2 

+ 

n−1 

D 

k=1 

2 k,T 

 

m 

T 

k + 1 

∗k T k 

k=0 

(−1) k 

in L(H). 

in L(H). 

 

m m − 1 

− T 

k + 1 k + 1 

∗k T k 

in L(H). 

We now use the standard formula m m−1 m−1 k+1 = k+1 + k for binomial coefficients to conclude 

that 

m−1 

Σm − Σm−1 = 

k=0 

(−1) k 

 

m − 1 

T 

k 

∗k T k = D 2 m−1,T 

An easy induction argument now completes the proof of the lemma. ✷ 

in L(H).


Let T ∈ L(H) be an n-hypercontraction. We denote by Hn the Hilbert space H equipped with 

the equivalent norm 

x 2 n = 

n−1 

k=0 

(−1) k 

 

n T 

k + 1 

k x 2 =x 2 n−1 

+ Dk,T x 2 , x ∈ H. (3.3) 

The equality of these two expressions for the norm ·n in (3.3) follows by Lemma 3.1. We 

denote by In the inclusion map of H into Hn defined by Inx = x for x ∈ H. 

Lemma 3.2. Let T ∈ L(H) be an n-hypercontraction, and consider the inclusion map 

In : H → Hn defined by Inx = x for x ∈ H. Then the adjoint operator I ∗ n ∈ L(Hn, H) acts 

as 

I ∗ n x = 

 

n−1 

(−1) k 

k=0 

Proof. For x ∈ Hn and y ∈ H we have that 

∗ 

In x,y n−1 

=〈x,Iny〉n = 

k=0 

(−1) k 

This gives the conclusion of the lemma. ✷ 

k=1 

 

n 

T 

k + 1 

∗k T k 

 

x, x ∈ Hn. 

 

n T n−1 

k k 

x,T y = 

k + 1 

We shall consider also the operator Tn = InT in L(H, Hn). 

k=0 

(−1) k 

 

n 

T 

k + 1 

∗k T k 

 

x,y . 

Lemma 3.3. Let T ∈ L(H) be an n-hypercontraction. Then the operator Tn = InT in L(H, Hn) 

is a contraction operator with defect operator and defect space given by 

The adjoint operator T ∗ n ∈ L(Hn, H) acts as 

Proof. Notice first that 

DTn = Dn,T in L(H) and DTn = Dn,T . 

T ∗ ∗ 

n x = T 

 

n−1 

(−1) k 

k=0 

I ∗ n In 

n−1 

= I + 

 

n 

T 

k + 1 

∗k T k 

 

x, x ∈ Hn. 

k=1 

D 2 k,T 

in L(H) 

by Lemmas 3.1 and 3.2. By the standard formula k+1 k k 

j = j + j−1 for binomial coefficients 

we have the equalities 

D 2 k+1,T = D2 k,T − T ∗ D 2 k,T T in L(H)


for 1 k n − 1. Using these equalities we compute that 

T ∗ n Tn = T ∗ I ∗ n InT = T ∗ n−1 

T + T ∗ D 2 k,T T = T ∗ n−1 

2 

T + Dk,T − D 2 

k+1,T 

k=1 

= T ∗ T + D 2 1,T − D2 n,T = I − D2 n,T 

in L(H). 

The equality T ∗ n Tn + D2 n,T = I in L(H) shows that the operator Tn ∈ L(H, Hn) is a contraction 

with defect operator and defect space as in the lemma. 

The action of the adjoint operator T ∗ n is evident by Lemma 3.2. ✷ 

We can now refine the description of the wandering subspace from Theorem 2.1. 

Theorem 3.1. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Let the operators Tn and 

In in L(H, Hn) be as above. Then a function f in An(Dn,T ) belongs to the wandering subspace 

En,T for In,T if and only if it has the form 

k=1 

f =−T ∗ n y + S′ −1 

nVnIn DT ∗ n y, y ∈ DT ∗ n . 

Furthermore, we have the norm equality that f 2 An =y2 n . 

Proof. By Theorem 2.1 we know that f ∈ An(Dn,T ) belongs to the wandering subspace En,T 

for In,T if and only if it has the form 

f = a0 + S ′ n Vnx 

for some elements a0 ∈ Dn,T and x ∈ H such that Eq. (2.2) holds. Using Lemma 3.3 we can 

rewrite Eq. (2.2) as 

using the operators Tn and In. Herea0∈Dn,T = DTn and x ∈ H. 

Let us now solve Eq. (3.4). We shall use the unitary operator 

and its adjoint operator 

θTn = 

θ ∗ Tn = 

 

T ∗ 

n 

Tn DT ∗ n 

T ∗ n Inx + DTn a0 = 0 (3.4) 

DTn −T ∗ n 

DT ∗ n 

a0 

 

: H ⊕ DT ∗ n → Hn ⊕ DTn 

DTn 

−Tn 

 

: Hn ⊕ DTn → H ⊕ DT ∗ n . 

Assume now that x ∈ H and a0 ∈ DTn satisfies (3.4). Then 

θ ∗ 

Inx T ∗ 

Tn = n Inx + DTna0 

0 

= , 

y 

DT ∗ n Inx − Tna0


where y = DT ∗ n Inx − Tna0 ∈ DT ∗. Applying the operator θTn to this last equality we find that 

n 

Inx 

a0 

 

= θTn 

 

0 DT 

= 

y 

∗ n y 

−T ∗ n y 

 

. 

This makes evident that every solution x ∈ H and a0 ∈ DTn of Eq. (3.4) is of the form 

 

x = I −1 

n DT ∗ n y, 

a0 =−T ∗ n y 

for some element y ∈ DT ∗ n . Also, if x ∈ H and a0 ∈ DTn are given by (3.5) for some element 

y ∈ DT ∗, then, by property (3.2) of defect operators, Eq. (3.4) holds. We have thus shown that 

n 

the solutions of (3.4) are parametrized by (3.5). By (3.5) we now have that 

f = a0 + S ′ nVnx =−T ∗ n y + S′ −1 

nVnIn DT ∗ n y, 

where y ∈ DT ∗ n . 

Let us now prove the norm equality that f 2 An =y2 n .Letx ∈ H and a0 ∈ Dn,T be given 

by (3.5). By Theorem 2.2 we have that 

f 2 An =a0 2 n−1 

+ 

k=0 

(−1) k 

 

n T 

k + 1 

k x 2 =a0 2 +x 2 n = T ∗ n y 2 +DT ∗ n y2 n =y2 n , 

where the last equality follows by (3.1). This completes the proof of the theorem. ✷ 

Let T ∈ L(H) be an n-hypercontraction. Notice that by Lemma 3.2 the operator TnT ∗ n in 

L(Hn) acts as 

TnT ∗ n x = TT∗ 

 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

 

x, x ∈ Hn. 

Since the operator Tn ∈ L(H, Hn) is a contraction by Lemma 3.3, the operator 

TT ∗ 

 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

 

in L(H) 

is self-adjoint in L(Hn) and has its spectrum contained in the closed unit interval [0, 1]. We 

denote by Qn,T the operator 

Qn,T = 

 

I − TT ∗ 

 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

1/2 in L(H), 

(3.5)


where the positive square root is computed in L(Hn). We denote by D∗ n,T the closure in H of 

the range of the operator Qn,T , that is, D∗ n,T = Qn,T (H), and we equip this space D∗ n,T with the 

Hilbert space structure given by the norm ·n defined by (3.3). 

We can now restate Theorem 3.1 using the space D∗ n,T as follows. 



f =−T ∗ 

 

n−1 

(−1) k 

k=0 

Furthermore, we have the norm equality that f 2 An =x2 n . 

 

n 

T 

k + 1 

∗k T k 

 

x + S ′ nVnQn,T x, x ∈ D ∗ n,T . (3.6) 

Proof. Recall the action of the adjoint operator T ∗ n ∈ L(Hn, H) given by Lemma 3.3. The map 

In : H → Hn naturally identifies the space D∗ n,T with the defect space DT ∗. The result is evident 

n 

by Theorem 3.1. ✷ 

We record also the following lemma. 

Lemma 3.4. Let T ∈ L(H) be an n-hypercontraction. Then 

in L(H). 

T ∗ 

 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

 

Qn,T = Dn,T T ∗ 

 

n−1 

k=0 

(−1) k 

 

n 

T 

k + 1 

∗k T k 

 

Proof. By (3.2) we have the formula T ∗ n DT ∗ n = DTnT ∗ n in L(Hn, H). Recall that DTn = Dn,T by 

Lemma 3.3, and the action of T ∗ n given by the same lemma. This makes evident the conclusion 

of the lemma. ✷ 

Let T ∈ L(H) be an n-hypercontraction. We recall from the introduction the definition of the 

operator-valued analytic function Wn,T : 

Wn,T (z) = 

z ∈ D. 

 

−T ∗ 

 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

 

n 

+ zDn,T (I − zT ) 

k=1 

−k 

 

D 

Qn,T , 

∗ 

n,T 

By Lemma 3.4 the values Wn,T (z) attained by this function Wn,T are operators in L(D ∗ n,T , Dn,T ). 

Notice that 

 

1 

μn;k+1 

k0 

z k = 1 

 

 

1 

− 1 

z (1 − z) n 

= 

n 

k=1 

1 

, z∈D. (1 − z) k


The function Wn,T thus has the power series expansion 

Wn,T (z) = 

 

−T ∗ 

 

n−1 

(−1) k 

k=0 

 

n 

T 

k + 1 

∗k T k 

 

+ 

k1 

1 

μn;k 

 

Dn,T T k−1 

D 

k 

Qn,T z , 

∗ 

n,T 

z ∈ D. (3.7) 

We can now parametrize the wandering subspace En,T for In,T using the function Wn,T as 

follows. 



f(z)= Wn,T (z)x, z ∈ D, (3.8) 

for some element x ∈ D∗ n,T . Furthermore, we have the norm equality 

when f is of the form (3.8). 

f 2 An =x2n =x2 n−1 

+ Dk,T x 2 , x ∈ D ∗ n,T , 

Proof. Let f ∈ En,T be given by (3.6). By formulas (0.5) and (1.2) we have that 

f(z)=−T ∗ 

 

n−1 

(−1) k 

k=0 

k=1 

 

n 

T 

k + 1 

∗k T k 

 

x + 

k1 

1 

μn;k 

By the power series expansion (3.7) of the function Wn,T we conclude that 

f(z)= Wn,T (z)x, z ∈ D. 

The conclusion of the theorem is now evident by Theorem 3.2. ✷ 

Dn,T T k−1 Qn,T x z k , z∈ D. 

We remark that in the case n = 1 of a contraction operator T ∈ L(H) the L(DT ∗, DT )-valued 

analytic function 

WT (z) = W1,T (z) = −T ∗ + zDT (I − zT ) −1 DT ∗ 

 

DT , z∈D, ∗ 

is the characteristic operator function studied by Sz.-Nagy and Foias (see [26, Chapter VI]). 

4. Multiplier properties of the function Wn,T 

In this section we discuss some multiplier properties of the function Wn,T .Wefirstshow 

that the function Wn,T acts as a contractive multiplier from the Hardy space A1(D∗ n,T ) into the 

Bergman space An(Dn,T ).


Theorem 4.1. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Then the function Wn,T 

acts as a contractive multiplier Wn,T : f ↦→ Wn,T f from the Hardy space A1(D∗ n,T ) into the 

Bergman space An(Dn,T ): 

Wn,T f 2 An f 2 ∗ 

A1 

, f ∈ A1 Dn,T ; 

here the space D ∗ n,T is equipped with the norm ·n given by (3.3). 

Proof. Let f ∈ A1(D∗ n,T ) be a polynomial of the form (0.1) with coefficients ak ∈ D∗ n,T .We 

write the function Wn,T f as 

Wn,T f = 

S k nWn,T ak. 

k0 

Recall that by Theorem 3.3 the elements Wn,T ak all belong to the wandering subspace En,T .We 

rewrite the sum for Wn,T f as 

 

 

f = Wn,T a0 + Sn S k nWn,T 

ak+1 . 

Now, since the wandering subspace En,T for In,T is orthogonal to Sn(In,T ), we have that 

k0 

Wn,T f 2 An =Wn,T a0 2 An + 

 

 

 

=a0 2 n + 

 

 

 

Sn 

Sn 

 

k0 

 

k0 

S k n Wn,T ak+1 


where the last equality follows by Theorem 3.3. The fact that the shift operator Sn on An(Dn,T ) 

is a contraction now gives that 

Wn,T f 2 An a0 2 n + 

 

 

 

 

k0 


We can now iterate this last inequality (4.1) to obtain that 

 

ak 2 n . 

Wn,T f 2 An 

k0 

 

2 

 

 

 

 

An 

2 

An 

, 

 

2 

An 

. (4.1) 

Since the space of D∗ n,T -valued polynomials is dense in A1(D∗ n,T ), an approximation argument 

now yields the conclusion of the theorem. ✷ 

Remark 4.1. We remark that the closure in An(Dn,T ) of the range of the multiplier 

Wn,T : A1(D ∗ n,T ) → An(Dn,T ), that is, the closure in An(Dn,T ) of the set of all functions of 

the form Wn,T g, where g ∈ A1(D∗ n,T ) is a D∗ n,T -valued polynomial, equals the shift invariant 

subspace [En,T ] generated by the wandering subspace En,T . In particular, the multiplier Wn,T 

maps A1(D ∗ n,T ) into In,T .


We recall that the space D∗ n,T is equipped with the norm ·n given by (3.3). In particular, 

this means that the norm of A1(D∗ n,T ) is given by 

f 2 

A1 

= 

k0 

for f ∈ A1(D∗ n,T ) as in (0.1). 

Let us consider the case n = 1 in some more detail. 

ak 2 n 

Corollary 4.1. Let T ∈ L(H) be a contraction in the class C0·. Then the characteristic operator 

function WT = W1,T is an isometric multiplier WT : f ↦→ WT f from the Hardy space A1(DT ∗) 

into the Hardy space A1(DT ) with range equal to I1,T . 

Proof. In this case the shift operator S1 on A1(DT ) is an isometry and we have equality 

in (4.1). This gives that the multiplier WT maps A1(DT ∗) isometrically into A1(DT ). Bythe 

von Neumann–Wold decomposition of an isometry (see [26, Section I.1]), the range of the multiplier 

WT equals I1,T (see Remark 4.1). ✷ 

We remark that the proof of Theorem 4.1 is modeled on an argument of Shimorin [25, 

Lemma 2.1]. 

Remark 4.2. In the scalar case when the defect spaces Dn,T and D∗ n,T are both one-dimensional 

and n = 2 the result of Theorem 4.1 is due to Hedenmalm [13,15]. The case n = 3 goes back to 

Hedenmalm [14]. 

We next show that the multiplier Wn,T : A1(D ∗ n,T ) → An(Dn,T ) is injective. 

Proposition 4.1. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Let the function Wn,T 

act as a multiplier from A1(D ∗ n,T ) into An(Dn,T ) as in Theorem 4.1. Denote by L the operator 

Then the intertwining relations 

L = (Sn|In,T )∗ Sn|In,T 

−1(Sn|In,T )∗ 

in L(In,T ). 

SnWn,T = Wn,T S1 and LWn,T = Wn,T L1 

holds. In particular, the multiplier Wn,T : A1(D ∗ n,T ) → An(Dn,T ) is injective. 

Proof. The first intertwining relation SnWn,T = Wn,T S1 is obvious. Let us verify the second 

intertwining relation LWn,T = Wn,T L1. Recall from Section 1 that the operator L in L(In,T ) is 

the left-inverse of Sn|In,T with kernel ker L = ker(Sn|In,T )∗ = En,T .Let 

g(z) = 

bkz k , z∈D, (4.2) 

k0


be a D∗ n,T -valued polynomial. The function f = Wn,T g has the form 

f = 

k0 

S k n Wn,T bk 

and the elements Wn,T bk all belong to En,T (see Theorem 3.3). We now have that 

Lf = 

k1 

S k−1 

n Wn,T bk = 

S k nWn,T bk+1 = Wn,T L1g. 

k0 

This shows that LWn,T g = Wn,T L1g when is g is a D∗ n,T -valued polynomial. The intertwining 

relation LWn,T = Wn,T L1 now follows by a standard approximation argument. 

Let us now prove that the multiplier Wn,T : A1(D∗ n,T ) → An(Dn,T ) is injective. We shall 

use the operator P = I − SnL in L(In,T ) which is the orthogonal projection of In,T onto En,T 

(see Section 1). Let f = Wn,T g, where g ∈ A1(D∗ n,T ) is given by (4.2). A computation using the 

intertwining relations shows that 

PL k f = (I − SnL)L k 

Wn,T g = Wn,T I − S1L 1 k 

1 L1g = Wn,T bk, k0. By Theorem 3.3 this last equality determines the coefficients bk uniquely. This completes the 

proof of the proposition. ✷ 

Recall that the reproducing kernel for a Hilbert space H of E-valued analytic functions in the 

unit disc D is the function KH : D × D → L(E) satisfying the conditions that KH(·,ζ)xbelongs to H for every ζ ∈ D and x ∈ E, and that 

 

f(ζ),x= f,KH(·,ζ)x 

H , ζ ∈ D, f∈ H, (4.3) 

for x ∈ E. This last property (4.3) is called the reproducing property of the kernel function KH. 

We remind that the Bergman space An(E) has the reproducing kernel 

Kn(z, ζ ) = 

where IE denotes the identity operator on E. 

We next compute the reproducing kernel for the space Vn(H). 

1 

(1 − ¯ζz) n IE, (z,ζ)∈ D × D, (4.4) 

Proposition 4.2. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Then the reproducing 

kernel for the space Vn(H) is the Dn,T -valued function given by 

KVn(H)(z, ζ ) = Dn,T (I − zT ) −n (I − ¯ζT ∗ ) −n Dn,T , (z,ζ)∈ D × D. 

Proof. Let f = Vnx be a function in Vn(H). Then for y ∈ Dn,T we have that 

 

f(ζ),y= Dn,T (I − ζT) −n x,y = x,(I − ¯ζT ∗ ) −n Dn,T y .


The fact that the operator Vn in L(H,An(Dn,T )) is an isometry now gives that 

 

f(ζ),y= Vnx,Vn(I − ¯ζT ∗ ) −n Dn,T y 

An = f,Vn(I − ¯ζT ∗ ) −n Dn,T y 

This completes the proof of the proposition. ✷ 

We denote by W the range of the multiplier Wn,T : g ↦→ Wn,T g mapping A1(D∗ n,T ) into 

An(Dn,T ) by Theorem 4.1, that is, the space W consists of all functions f ∈ An(Dn,T ) of the 

form 

f(z)= Wn,T (z)g(z), z ∈ D, (4.5) 

for some g ∈ A1(D∗ n,T ). Recall that by Proposition 4.1 the function f ∈ W uniquely determines 

the function g ∈ A1(D∗ n,T ) by (4.5). We equip the space W with the norm induced by A1(D∗ n,T ), 

that is, we set f 2 W =g2 A1 when f ∈ W and g ∈ A1(D∗ n,T ) are related as in (4.5). In this way 

the space W becomes a Hilbert space of Dn,T -valued analytic functions in D. We next compute 

the reproducing kernel function for the space W. 

Lemma 4.1. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Let the Hilbert space 

W of Dn,T -valued analytic functions in D be defined as in the previous paragraph. Then the 

reproducing kernel for the space W is given by 

KW(z, ζ ) = 1 

1 − ¯ζz Wn,T (z)Wn,T (ζ ) ∗ , (z,ζ)∈ D × D. 

Proof. Let f ∈ W and g ∈ A1(D ∗ n,T ) be as in (4.5). For x ∈ Dn,T we have that 

f(ζ),x = Wn,T (ζ )g(ζ ), x = g(ζ),Wn,T (ζ ) ∗ x 

n . 

By the reproducing property of the kernel function K1 for the space A1(D∗ n,T ) we have that 

 

f(ζ),x= g,K1(·,ζ)Wn,T (ζ ) ∗ x 

Now since the function Wn,T acts as an isometric multiplier of A1(D∗ n,T ) onto the space W we 

conclude that 

f(ζ),x = Wn,T g,Wn,T K1(·,ζ)Wn,T (ζ ) ∗ x 

W = f,Wn,T K1(·,ζ)Wn,T (ζ ) ∗ x 

W . 

This completes the proof of the lemma. ✷ 

The contractive multiplier property from Theorem 4.1 leads to the following result on domination 

of reproducing kernel functions. 

Theorem 4.2. Let T ∈ L(H) be an n-hypercontraction in the class C0·. Then the Dn,T -valued 

function 

A1 . 

An .


L(z, ζ ) = 

1 

(1 − ¯ζz) n IDn,T − Dn,T (I − zT ) −n (I − ¯ζT ∗ ) −n Dn,T 

− 1 

1 − ¯ζz Wn,T (z)Wn,T (ζ ) ∗ , (z,ζ)∈ D × D, 

is positive definite on D × D. In particular, we have the inequality 

1 

1 −|z| 2 Wn,T (z)Wn,T (z) ∗ + Dn,T (I − zT ) −n (I −¯zT ∗ ) −n Dn,T 

 

1 

(1 −|z| 2 ) n IDn,T in L(Dn,T ), z ∈ D. 

Proof. Let the space W be as in Lemma 4.1. By Theorem 4.1 and Remark 4.1 the space W 

is contractively embedded into In,T . By this we mean that W ⊂ In,T and f 2 An f 2 W for 

f ∈ W. Recall that the space An(Dn,T ) is the orthogonal sum of the subspaces Vn(H) and In,T . 

The reproducing kernel function for the space In,T is given by 

KIn,T (z, ζ ) = Kn(z, ζ ) − KVn(H)(z, ζ ) 

= 

1 

(1 − ¯ζz) n IDn,T − Dn,T (I − zT ) −n (I − ¯ζT ∗ ) −n Dn,T , (z,ζ)∈ D × D, 

where the last equality follows by Proposition 4.2 and (4.4). The reproducing kernel function 

KW for the space W was computed in Lemma 4.1. It is known that a contractive embedding 

W ⊂ In,T is equivalent to the domination relation KW ≪ KIn,T of reproducing kernel functions 

(see [7, Section I.7]). We conclude that the function 

L(z, ζ ) = KIn,T (z, ζ ) − KW(z, ζ ), (z, ζ ) ∈ D × D, 

is positive definite on D × D. This gives the positive definiteness assertion in the theorem. The 

last inequality in the theorem follows by setting z = ζ noticing that L(z, z) 0inL(Dn,T ) by 

positive definiteness of the function L. This completes the proof of the theorem. ✷ 

In the case n = 2 of scalar-valued Bergman inner functions the inequality in Theorem 4.2 

seems first to have appeared in Zhu [27, Theorem 4.2]. 

Let us examine the case n = 1 somewhat closer. 

Corollary 4.2. Let T ∈ L(H) be a contraction in the class C0·. Then 

1 

1 

IDT = 

1 − ¯ζz 1 − ¯ζz WT (z)WT (ζ ) ∗ + DT (I − zT ) −1 (I − ¯ζT ∗ ) −1 DT , 

(z, ζ ) ∈ D × D. 

Proof. The space A1(DT ) is the orthogonal sum of the subspaces V1(H) and I1,T . By Corollary 

4.1 we know that the characteristic operator function WT is an isometric multiplier from 

A1(DT ∗) onto I1,T . The reproducing kernel functions decompose accordingly. ✷


We remark that the result of Corollary 4.2 is a well-known formula in the context of unitary 

systems which can be proved by direct computation (see [10, Section XXVIII.2]). We have 

included it here merely as an illustration of our methods. 

5. Shift invariant subspaces in Bergman spaces 

The considerations in the previous sections have some consequences concerning general shift 

invariant subspaces in Bergman spaces. Let I be a shift invariant subspace of An(E). Weset 

H = An(E) ⊖ I and T = S ∗ n |H. 

The operator T in L(H) is an n-hypercontraction in the class C0·. We can now model this operator 

T as part of the adjoint shift operator S∗ n on the space An(Dn,T ) by means of the formula 

(Vnf )(z) = Dn,T (I − zT ) −n f = 

 

k + n − 1 Dn,T T 

k 

k f z k , z∈D. k0 

Furthermore, by a uniqueness property of this operator model there exists an isometry ˆVn : 

Dn,T → E such that the functions f ∈ H all admit the representation 

 

f(z)= ˆVn (Vnf )(z) , z∈D. (5.1) 

The isometry ˆVn : Dn,T → E of coefficient spaces is uniquely determined by (5.1) and acts as 

ˆVn : Dn,T f ↦→ f(0) for f ∈ H. 

Full details of this construction can be found in [22, Sections 6 and 7]. 

We write Ê = ˆVn(Dn,T ) ⊂ E. ThemapˆVn naturally extends to an isometry of An(Dn,T ) into 

An(E) with range equal to An(Ê) by setting 

 

( ˆVnf )(z) = ˆVn f(z) , z∈D, for f ∈ An(Dn,T ). 

We have the following description of a general shift invariant subspace of An(E). 

Theorem 5.1. Let I be a shift invariant subspace of An(E), and let H = An(E) ⊖ I and T = 

S ∗ n |H in L(H) be as above. Then the space I decomposes as an orthogonal sum 

I = An(E ⊖ Ê) ⊕ ˆVn(In,T ), 

that is, a function f ∈ An(E) belongs to I if and only if it has the form of an orthogonal sum 

f = f1 + ˆVng, where f1 ∈ An(E) has all its Taylor coefficients in E ⊖ Ê and g belongs to the 

shift invariant subspace In,T of An(Dn,T ). 

Proof. By formula (5.1) we have that ˆVn(Vn(H)) = H. In particular, the space H is contained 

already in An(Ê). Passing to the orthogonal complement we have that


This completes the proof of the theorem. ✷ 

I = An(E) ⊖ H = An(E ⊖ Ê) ⊕ An(Ê) ⊖ H 

 

= An(E ⊖ Ê) ⊕ ˆVn An(Dn,T ) ⊖ Vn(H) 

= An(E ⊖ Ê) ⊕ ˆVn(In,T ). 

Recall that the wandering subspace EI for a shift invariant subspace I of An(E) is the subspace 

EI = I ⊖ Sn(I) 

of I. We have the following description of a general wandering subspace. 

Theorem 5.2. Let I be a shift invariant subspace of An(E), and let H = An(E) ⊖ I and T = 

S ∗ n |H in L(H) be as above. Then a function f ∈ An(E) belongs to the wandering subspace EI 

for I if and only if it has the form 

f(z)= a0 + ˆVnWn,T (z)g, z ∈ D, (5.2) 

where a0 ∈ E ⊖ Ê and g belongs to the defect space D ∗ n,T . The norm of a function f ∈ An(E) of 

the form (5.2) is given by 

f 2 An =a0 2 +g 2 n =a0 2 + 

where bk is the kth Taylor coefficient of g as in (4.2). 

μ 2 n;k 

μn;k+1 

k0 

bk 2 , 

Proof. The form of the invariant subspace I was calculated in Theorem 5.1. By this description 

we have that 

EI = (E ⊖ Ê) ⊕ ˆVn(En,T ), 

where En,T is the wandering subspace for the shift invariant subspace In,T of An(Dn,T ). The 

wandering subspace En,T was described in Theorem 3.3 as the space of all functions of the form 

f(z)= Wn,T (z)g, z ∈ D, 

where g belongs to D∗ n,T , and norm given by f 2 An =g2n . The expression for the norm ·n 

using Taylor coefficients follows by Proposition 1.1. This yields the conclusion of the theorem. 

✷ 

We have the following characterization of a related class of operator-valued analytic functions. 

Theorem 5.3. Let W be an L(F, E)-valued analytic function in the unit disc D such that for 

every x ∈ F the function Wx : z ↦→ W(z)x belongs to An(E) with the properties that


• the norm equality Wx2 An =x2 holds for every x ∈ F, and 

• Wx ⊥ Sk nWx for all k 1 and x ∈ F. 

Then W(F) ={Wx: x ∈ F} is a wandering subspace in An(E). Denote by I the shift invariant 

subspace generated by W(F) in An(E), that is, I =[W(F)]. Let H = An(E) ⊖ I and T = 

S ∗ n |H ∈ L(H). Then there exists a unitary operator 

of F onto (E ⊖ Ê) ⊕ D ∗ n,T 

U = 

such that 

U1 

U2 

 

: F → (E ⊖ Ê) ⊕ D ∗ n,T 

W(z)x = U1x + ˆVnWn,T (z)U2x, x ∈ F, z∈ D. (5.3) 

Furthermore, the equality (5.3) determines the operators U1 and U2 uniquely. 

Proof. Clearly W(F) is a closed subspace of An(E). By polarization we have that Wx ⊥ Sk nWy for k 1 and x,y ∈ F. This shows that W(F) satisfies the defining property of a wandering 

subspace, that is, W(F) ⊥ Sk nW(F) for k 1. 

We have that EI = W(F) (see Remark 5.1). By Theorem 5.2 the wandering subspace EI = 

W(F) for I consists of all functions f in An(E) of the form (5.2) with norm equality 

f 2 An =a0 2 +g 2 n . 

For x ∈ F we set Ux = (a0,g) when f = Wx is given by (5.2). This gives us a unitary map 

U from F onto (E ⊖ Ê) ⊕ D ∗ n,T . The operators U1 and U2 are uniquely determined by (5.3) by 

uniqueness of the representation (5.2). This completes the proof of the theorem. ✷ 

Remark 5.1. It is a general fact often accredited to Halmos [12] that a wandering subspace is 

uniquely determined by the invariant subspace it generates. Let T ∈ L(H) be a bounded linear 

operator, and let E be a closed subspace of H such that E ⊥ T k (E) for k 1. Set I =[E]T 

= 

k0 T k (E). Then I = E ⊕ ( 

k1 T k (E)), which gives that EI = I ⊖ T(I) = E. 

We recall that the operator-valued analytic function Wn,T is a contractive multiplier from the 

Hardy space A1(D ∗ n,T ) into the Bergman space An(Dn,T ) (see Theorem 4.1), and that a related 

upper bound is available (see Theorem 4.2). 

Specializing to the case n = 1 we obtain a parametrization of the shift invariant subspaces of 

the Hardy space A1(E) for a general not necessarily separable Hilbert space E. 

Corollary 5.1. Let I be a shift invariant subspace of the Hardy space A1(E). Then a function f 

in A1(E) belongs to the subspace I if and only if it has the form 

f(z)= f1(z) + ˆV1WT (z)g(z), z ∈ D, (5.4) 

for some functions f1 ∈ A1(E ⊖ Ê) and g ∈ A1(DT ∗). Furthermore, we have the norm equality 

f 2 A1 =f12 A1 +g2 A1 when f ∈ I, f1 ∈ A1(E ⊖ Ê) and g ∈ A1(DT ∗) are related by (5.4).


Proof. Recall the form of an invariant subspace calculated in Theorem 5.1. By Corollary 4.1 the 

characteristic operator function WT is an isometric multiplier from the Hardy space A1(DT ∗) into 

the Hardy space A1(DT ) with range equal to I1,T . This completes the proof of the corollary. ✷ 

The previous results yield the following consequence concerning the index dim EI of a shift 

invariant I of An(E). 

Corollary 5.2. Let I be a shift invariant subspace of An(E) with E separable. Set H = An(E)⊖I 

and T = S ∗ n |H. Then dim Dn,T dim E. If the defect index dim Dn,T is finite, then 

dim EI = dim E − dim Dn,T + dim D ∗ n,T . 

Proof. The first inequality dim Dn,T dim E is evident by the fact that the operator ˆVn ∈ 

L(Dn,T , E) is an isometry (see [22, Section 7]). The second inequality is evident by the description 

of the wandering subspace EI in Theorem 5.2. ✷ 

We notice also that dim EI = dim D ∗ n,T if ˆVn(Dn,T ) = E. 

It has been known for some time that even in the scalar case E = C the index dim EI of a shift 

invariant subspace I of An(C) for n 2 can equal any positive integer or +∞. This was first 

proved by Apostol et al. [5] using dual algebras, and later more explicit constructions have been 

found by Hedenmalm et al. [19] and others. 

In the context of the Hardy space A1(E) with E separable it is a result of Halmos [12] that the 

index dim EI of a shift invariant subspace I of A1(E) cannot exceed the index of the whole space 

A1(E) meaning that dim EI dim E. This inequality is naturally interpreted as an inequality of 

defect indexes as follows. 

Proposition 5.1. Let T ∈ L(H) be a contraction operator in the class C0· acting on a separable 

Hilbert space H. Then dim DT ∗ dim DT . 

Proof. It is known that the characteristic operator function WT has non-tangential boundary values 

WT (e iθ ) in the strong operator topology for a.e. e iθ ∈ T. A well-known argument then shows 

that the operator WT (e iθ ) is an isometry in L(DT ∗, DT ) for a.e. e iθ ∈ T (see [26, Chapter V]). 

This gives the conclusion of the proposition. ✷ 

For an n-hypercontraction T ∈ L(H) in the class C0· and n 2 the corresponding inequality 

dim D ∗ n,T dim Dn,T of defect indexes is not true in general for the reasons quoted above. 

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Hua operators and Poisson transform for bounded 

symmetric domains ✩ 

Khalid Koufany a , Genkai Zhang b,∗ 

a Institut Élie Cartan, UMR 7502, Université Henri Poincaré (Nancy 1), BP 239, 

F-54506 Vandœuvre-lès-Nancy cedex, France 

b Department of Mathematics, Chalmers University of Technology and Göteborg University, 

S-41296 Göteborg, Sweden 

Received 8 December 2005; accepted 24 February 2006 


Communicated by Paul Malliavin 

Abstract 

Let Ω be a bounded symmetric domain of non-tube type in Cn with rank r and S its Shilov boundary. We 

consider the Poisson transform Psf(z)for a hyperfunction f on S defined by the Poisson kernel Ps(z, u) = 

(h(z, z) n/r /|h(z, u) n/r | 2 ) s , (z, u)×Ω ×S, s ∈ C.Forallssatisfying certain non-integral condition we find 

a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua 

operators. When Ω is the type I matrix domain in Mn,m(C) (n m), we prove that an eigenvalue equation 

for the second order Mn,n-valued Hua operator characterizes the image. 


Keywords: Bounded symmetric domains; Shilov boundary; Invariant differential operators; Eigenfunctions; Poisson 

transform; Hua systems 


Let Ω = G/K be a Riemannian symmetric space. Any parabolic subgroup P of G defines a 

boundary G/P of the symmetric space Ω. The Poisson transform is an integral operator from 

hyperfunctions on G/P into the space of eigenfunctions on Ω of the algebra D(Ω) G of invariant 

✩ Research of the authors is partly supported by European IHP network Harmonic Analysis and Related Problems. 

Research of G. Zhang is supported by Swedish Science Council (VR). 

* Corresponding author. 

E-mail addresses: khalid.koufany@iecn.u-nancy.fr (K. Koufany), genkai@math.chalmers.se (G. Zhang). 


doi:10.1016/j.jfa.2006.02.014

K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580 547 

differential operators. Any such boundary G/P can be viewed as coset space of the maximal 

boundary G/Pmin defined by a minimal parabolic subgroup Pmin. In this case the most general 

result was obtained by Kashiwara et al. [7] where they proved that under certain conditions on 

the eigenvalues the Poisson transform is a G-isomorphism between the space of hyperfunctions 

on G/Pmin and the space of eigenfunctions of invariant differential operators on Ω, namely 

the Helgason conjecture. It thus arises the question of characterizing the image of the Poisson 

transform for other smaller boundaries. 

Suppose Ω is a bounded symmetric domain in a complex n-dimensional space V .LetSbe its Shilov boundary and r its rank. In this paper we consider the characterization of the image of 

the Poisson transform 

 

Psϕ(z) = Ps(z, u)ϕ(u) dσ (u) 

on the Shilov boundary S when s satisfies the following condition: 

 

−4 b + 1 + j a 

2 

S 

 

n 

+ (s − 1) /∈{1, 2, 3,...} for j = 0 and 1, (1) 

r 

where a and b are some structure constants of Ω. For a specific value of s (s = 1 in our parameterization) 

the kernel Ps(z, u), (z, u) ∈ Ω × S, is the so-called Poisson kernel for harmonic 

functions, and the corresponding Poisson transform P := P1 maps hyperfunctions on S to harmonic 

functions on Ω; here harmonic functions are defined as the smooth functions that are 

annihilated by all invariant differential operators that annihilate the constant functions. When Ω 

is a tube domain Johnson and Korányi [6] proved that the image of the Poisson transform P is 

exactly the set of all Hua-harmonic functions. For non-tube domains the characterization of the 

image of the Poisson transform P was done by Berline and Vergne [1] where certain third-order 

differential Hua operator was introduced to characterize the image. 

In his paper [12] Shimeno considered the Poisson transform Ps on tube domains; it is proved 

that Poisson transform maps hyperfunctions on the Shilov boundary to certain solution space 

of the Hua operator. For general domains and for other boundaries, the image of the Poisson 

transform was characterized in [13]. However for the Shilov boundary of a non-tube domain 

the problem is still open. We will construct two Hua operators of the third order and use them 

to give a characterization. For the matrix ball Ir,r+b of r × (r + b)-matrices some eigenvalue 

equation for the second-order Hua operator (constructed by Hua [5] and reformulated by Berline 

and Vergne [1]) is proved to give the characterization. We proceed to explain the content of our 

paper. 

Hua operator of the second-order H for a general symmetric domain is defined as a kC-valued 

operator, see Section 4. For tube domains it maps the Poisson kernels into the center of kC, 

namely the Poisson kernels are its eigenfunctions up to an element in the center, but it is not 

true for non-tube domains, see Section 5. However for type I domains of non-tube type, see 

Section 6, there is a variant of the Hua operator, H (1) , by taking the first component of the 

operator, since in this case kC = k (1) 

C + k(2) 

C is a sum of two irreducible ideals. We prove that 

operator H (1) has the Poisson kernels as its eigenfunctions and we find the eigenvalues. We 

prove further that the eigenfunctions of the Hua operator H (1) are also eigenfunctions of invariant 

differential operators on Ω. For that purpose we compute the radial part of Hua operator H (1) ,

548 K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580 

see Proposition 6.3. We give eventually the characterization of the image of Poisson transform 

in terms of Hua operator for type Ir,r+b domains: 

Theorem 1.1 (Theorem 6.1). Suppose s ∈ C satisfies the following condition: 

−4 b + 1 + j + (r + b)(s − 1) /∈{1, 2, 3,...} for j = 0 and 1. 

A smooth function f on Ir,r+b is the Poisson transform Ps(ϕ) of a hyperfunction ϕ on S if and 

only if 

H (1) f = (r + b) 2 s(s − 1)f Ir. 

Our method of proving the characterization is the same as that in [8] by proving that the boundary 

value of the Hua eigenfunctions satisfy certain differential equations and is thus defined only 

on the Shilov boundary, nevertheless it requires several technically demanding computations. In 

Section 7 we study the characterization of the range of Poisson transform for general non-tube 

domains. We construct two new Hua operators of the third order and prove, by essentially the 

same method as for the previous theorem, the characterization of the image of Poisson transform 

using the third-order Hua-type operators U and W: 

Theorem 1.2 (Theorem 7.2). Let Ω be a bounded symmetric non-tube domain of rank r in C n . 

Let s ∈ C and put σ = n r s. If a smooth function f on Ω is the Poisson transform Ps of a hyperfunction 

in B(S), then 

Conversely, suppose s satisfies the condition 

 

−4 b + 1 + j a 

2 

 

U − −2σ 2 + 2pσ + c 

σ(2σ − p − b) W 

 

f = 0. (2) 

 

n 

+ (s − 1) /∈{1, 2, 3,...} for j = 0 and 1. 

r 

Let f be an eigenfunction f ∈ M(λs) (see (8)) with λs given by (11). Iff satisfies (2) then it is 

the Poisson transform Ps(ϕ) of a hyperfunction ϕ on S. 

After this paper was finished we were informed by Professor T. Oshima that he and N. Shimeno 

have obtained some similar results about Poisson transforms and Hua operators. 

2. Preliminaries and notation 

2.1. General setting 

We recall some basic facts about the Jordan triple characterization of bounded symmetric 

domains and fix notations. Our presentation is mainly based on [9]. Let Ω be an irreducible 

bounded symmetric domain in a complex n-dimensional space V .LetG be the identity component 

of the group of biholomorphic automorphisms of Ω, and K be the isotropy subgroup of G


at the point 0 ∈ Ω. Then K is a maximal compact subgroup of G and as a Hermitian symmetric 

space, Ω = G/K. Letg be the Lie algebra of G, and 

g = k + p 

be its Cartan decomposition. The Lie algebra k of K has one-dimensional center z. Then there 

exists an element Z0 ∈ z such that ad Z0 defines the complex structure of p.Let 

gC = p + ⊕ kC ⊕ p − 

be the corresponding eigenspace decomposition of gC, the complexification of g. We will use the 

Jordan theoretic characterization of Ω; the corresponding Lie theoretic characterization will be 

then more transparent and which we will also use. 

There exists a quadratic map Q : V → End( ¯V,V) (here ¯V is the complex conjugate of V ), 

such that 

p ={ξv: v ∈ V }, 

where ξv(z) = v − Q(z) ¯v. We will hereafter identify p + with V by the natural mapping 

and p − with ¯V by the mapping 

1 

2 (ξv − iξiv) = v ↦→ v, 

− 1 

2 (ξv + iξiv) = Q(z) ¯v ↦→ ¯v ∈ ¯V ; 

we will write ¯v = Q(z) ¯v when viewed as element in the Lie algebra and when no ambiguity 

would arise. 

Let {z ¯vw} be the polarization of Q(z) ¯v, i.e. 

{z ¯vw}=Q(z + w)¯v − Q(z) ¯v − Q(w) ¯v. 

This defines a triple product V × ¯V × V → V , with respect to which V is a JB ⋆ -triple, see [17]. 

We define D(z, ¯v) ∈ End(V ) by 

D(z, ¯v)w ={z ¯vw}. 

The space V carries a K-invariant inner product 

〈z, w〉= 1 

tr D(z, ¯w), (4) 

p 

where “tr” is the trace functional on End(V ), and p = p(Ω) is the genus of Ω (see Eq. (6)). 

Beside the Euclidean norm, V carries also the spectral norm, 

 

 

z= 

1 

D(z, ¯z) 

2 

1/2 

, 

(3)


where the norm of an operator in End(V ) is taken with respect to the Hilbert norm 〈·,·〉 1/2 on V . 

The domain Ω can now be realized as the open unit ball of V with respect to the spectral norm, 

Ω = z ∈ V : z < 1 . 

An element c ∈ V is a tripotent if {c ¯cc}=c. In the matrix Cartan domains (of the type I, II, 

and III, see below) the tripotents are exactly the partial isometries. Each tripotent c ∈ V gives 

rise to a Peirce decomposition of V , 

where 

V = V0(c) ⊕ V1(c) ⊕ V2(c), 

Vj (c) = v ∈ V : D(c, ¯c)v = jv . 

Two tripotents c1 and c1 are orthogonal if D(c1, ¯c2) = 0. Orthogonality is a symmetric relation. 

A tripotent c is minimal if it cannot be written as a sum of two non-zero orthogonal tripotents. 

A tripotent c is maximal if V0(c) ={0}. AJordan frame is a maximal family of pairwise orthogonal, 

minimal tripotents. It is known that the group K acts transitively on Jordan frames. In 

particular, the cardinality of all Jordan frames is the same, and is equal to the rank r of Ω.Every 

z ∈ V admits a (unique) spectral decomposition z = r j=1 sj vj , where {vj } is a Jordan frame 

and s1 s2 ··· sr 0arethespectral values of z. The spectral norm of z is equal to the 

largest spectral value s1. 

Let us choose a Jordan frame {cj } r j=1 in V . Then, by the transitivity of K on frames, each 

element z ∈ V admits a polar decomposition z = k r j=1 sj cj , where k ∈ K and sj are the spectral 

values of z. Lete = c1 + c2 +···+cr; then e is a maximal tripotent and the G-orbit, G · e, 

is the Shilov boundary S of Ω.Let 

V = 

0jkr 

be the joint Peirce decomposition of V associated with the Jordan frame {cj } r j=1 , where 

Vj,k = v ∈ V : D(cℓ, ¯cℓ)v = (δℓ,j + δℓ,k)v: 1ℓ r 

for (j, k) = (0, 0) and V0,0 ={0}. By the minimality of cj , Vj,j = Ccj ,1 j r. The transitivity 

of K on the frames implies that the integers 

Vj,k 

a := dim Vj,k (1 j


V2 becomes a Jordan algebra for the product xy ={xēy} with identity element e.Letn1 = dim V1 

and n2 = dim V2. Then we have 

The genus of Ω is 

Thus 

n1 = rb, n2 = r + 

and this is true for every minimal tripotent in V . 

Let 

r(r − 1) 

a and n = n1 + n2. 

2 

p = p(Ω) = 1 

tr D(e,ē) = (r − 1)a + b + 2. (6) 

r 

〈cj ,cj 〉= 1 

p tr D(cj , ¯cj ) = 1 

tr D(e,ē) = 1, 

rp 

a = Rξ1 +···+Rξr, ξj = ξcj ,j= 1,...,r. 

Then, a is the maximal abelian subspace of p.Let{βj } r j=1 ⊂ a∗ be the basis of a ∗ determined by 

and define an ordering on a ∗ such that 

βj (ξk) = 2δj,k, 1 j,k r, 

βr >βr−1 > ···>β1 > 0. (7) 

The restricted roots system Σ(g, a) of g relative to a is of type Cr or BCr and it consists of 

the roots ±βj (1 j r) with multiplicity 1, the roots ± 1 2 βj ± 1 2 βk (1 j = k r) with 

multiplicity a, and possibly the roots ± 1 2 βj (1 j r) with multiplicity 2b. The set positive 

roots Σ + (g, a) consists of 1 2 (βk ± βj ) (1 j


The Iwasawa decomposition is then given by 

g = k ⊕ a ⊕ n − . 

Let, as usual, m = Zk(a) be the centralizer of a in k, then we have 

g = n − ⊕ m ⊕ a ⊕ n + . 

We let P = Pmin = MAN be the minimal parabolic subgroup of G, with M, A and N the 

corresponding Lie groups with Lie algebras m, a and n− . 

Let t − 

C be the subspace 

t − 

C = CD(c1, ¯c1) +···+CD(cr, ¯cr) 

of kC. Then t − 

C is abelian and we extend it to a Cartan subalgebra tC = t − 

C + t+ 

C of kC. The root 

system Ψ := Σ(gC, tC) of gC with respect to tC, when restricted to t − 

C is of the form 

Ψ | − 

tC = Σ(gC, t − 

 

C ) = ± 1 

2 (γk ± γj ), 1 j = k r; ±γj , ± 1 

2 γj 

 

, 1 j r , 

where γj are the Harish-Chandra strongly orthogonal roots defined by 

 

γj D(ck, ¯ck) = 2δjk, γj | + 

t = 0, 1 j,k r. 

C 

The set of compact roots Ψc := Σ(kC, tC) is such that 

Ψc| − 

tC = 

 

1 

2 (γk − γj ), 1 j = k r; ± 1 

2 γj 

 

, 1 j r , 

and the set of non-compact roots Ψn satisfies 

Ψn| − 

tC = 

 

± 1 

2 (γk + γj ), 1 j = k r; ±γj , ± 1 

2 γj 

 

, 1 j r . 

We choose a consistent ordering with (3) and (7) 

γr >γr−1 > ···>γ1. 

We will also need the set of positive non-compact roots Ψn| + 

, 

Ψn| + 

t − 

 

1 

= 

C 2 (γk + γj ), 1 j = k r; 1 

2 γj 

 

,γj , 1 j r . 

t − 

C


2.2. Bounded symmetric domain of type Ir,r+b 

Let V = Mr,r+b(C) be the vector space of complex r × (r + b)-matrices. V is a Jordan triple 

system for the following triple product: 

Then the endomorphisms D(z, ¯v) are given by 

{x ¯yz}=xy ∗ z + zy ∗ x. 

D(z, ¯v)w ={z ¯vw}=zv ∗ w + wv ∗ z. 

There is a canonical and natural choice of frames. One considers the standard matrix units {ei,j , 

1 i r, 1 j r + b} and defines cj = ej,j, 1 j r. Then the Pierce decomposition 

V = 

0jkr Vj,k of V is given by 

Let 

Vj,j = Ccj , 1 j r, 

Vj,k = Cej,k + Cek,j, 1 j


is easily seen to be a maximal compact subgroup of G. The Lie algebra g of G decomposes into 

g = k + p, where k, the Lie algebra of K, consists of all matrices 

 

a 0 

, a ∈ Mr,r(C), d ∈ Mr+b,r+b(C), a 

0 d 

∗ =−a, d ∗ =−d, 

and p consists of all matrices 

 

0 v 

v∗ 

, v∈Mr,r+b(C). 0 

The induced vector fields are given respectively by 

z ↦→ az − zd, and z ↦→ ξv(z) = v − zv ∗ z. 

The complex Lie algebra kC is given by the set of all matrices 

 

a 

0 

 

0 

, 

d 

a ∈ Mr,r(C), d ∈ Mr+b,r+b(C), tr(a) + tr(d) = 0. 

Hence, kC can be written as the sum 

where k (1) 

C 

and 

and k(2) 

C 

kC = k (1) 

C ⊕ k(2) 

C , 

are the ideals consisting respectively of the matrices 

 

a 0 

0 − tr(a) 

r+b Ir+b 

 

, a ∈ Mr,r(C), 

 

0 0 

, d ∈ Mr+b,r+b(C), tr(d) = 0. 

0 d 

Then, identifying kC as linear transformations of V ,wehave 

kC = span D(u, ¯v), u,v ∈ V 

and 

where the endomorphism D(u, ¯v) (1) is given by 

k (1) 

C = span D(u, ¯v) (1) ,u,v∈ V , 

D(u, ¯v) (1) z = uv ∗ z.

3. The Poisson transform 


Let D(Ω) G be the algebra of all invariant differential operators on Ω. Recall the definition 

of the Harish-Chandra eλ-function: eλ, forλ∈a∗ C is the unique N-invariant function on Ω such 

that 

 

eλ exp(t1ξ1 +···+trξr) · 0 = e 2t1(λ1+ρ1)+···+2tr (λr +ρr ) 

. 

Then eλ are the eigenfunctions of T ∈ D(Ω) G and we denote χλ(T ) the corresponding eigenvalues. 

Denote further 

M(λ) = f ∈ C ∞ (Ω): Tf = χλ(T )f, T ∈ D(Ω) G . (8) 

Recall the parabolic subgroup P = Pmin introduced in Section 2.1. Corresponding to P there 

is the Poisson transform on the maximal boundary G/P = K/M.Forλ∈a∗ C ,thePoisson transform 

Pλ,K/M is defined by 

 

Pλ,K/Mf(gK)= 

on the space B(K/M) of hyperfunctions on K/M. 

It is proved by Kashiwara et al. [7] that for λ ∈ a ∗ C ,if 

K 

−1 

eλ k g f(k)dk 

−2 〈λ,α〉 

/∈{1, 2, 3,...} (9) 

〈α, α〉 

for all α ∈ Σ + (g, a), then the Poisson transform is a G-isomorphism from B(K/M) onto M(λ). 

We now introduce the Poisson transform on the Shilov boundary. Let h(z) be the unique 

K-invariant polynomial on V whose restriction to Rc1 +···+Rcr is given by 

h 

r 

j=1 

tj cj 

 

= 

r 

j=1 

2 

1 − tj . 

As h is real-valued, we may polarize it to get a polynomial on V × V , denoted by h(z, w), 

holomorphic in z and antiholomorphic in w such that h(z, z) = h(z). Recall that the function h 

is related to the Bergman operator (see (12)) by that 

The Poisson kernel P(z,u)on Ω × S is 

det b(z, ¯z) = h(z, ¯z) p . 

 

h(z, z) 

P(z,u)= 

|h(z, u)| 2 

n/r .


For a complex number s we define the Poisson transform Psϕ on the space B(S) of hyperfunctions 

ϕ on S by 

 

(Psϕ)(z) = 

S 

P(z,u) s ϕ(u) dσ (u). 

The kernel P(z,u) s has the following transformation property: 

P(gz,gu) s = Jg(u) 2ns 

− rp s 

P(z,u) , ∀g ∈ G, (10) 

where Jg(u) is the Jacobian of g at u. 

The kernel P(z,u) s ,foru = e is a special case of the eλ-function. The Poisson transform Ps 

on S can be viewed as a restriction of the Poisson transform Pλ,K/M. However for fixed s there 

are various choices of λ and we will find a specific λ so that the above condition (9) is valued 

when s satisfies (1). Let 

and consider the decomposition 

ξc = ξ1 +···+ξr 

a = Rξc ⊕ ξ ⊥ c = Rξc 

r−1 

⊕ R(ξj − ξj+1) 

under the (negative) Killing form on g. We denote ξ ∗ c the dual vector, ξ ∗ c (ξc) = 1. We extend ξ ∗ c 

to a by the orthogonal projection defined above. Observe first that 

We have then 

j=1 

ρ(ξc) = n = nξ ∗ c (ξc). 

Psf(z)= Pλs,K/Mf(z), 

where f on S is viewed as a function on K and thus on K/M, λs ∈ a∗ C is given by 

Thus 

λs = ρ + 2n(s − 1)ξ ∗ c . (11) 

PsB(S) ⊂ Pλs,K/MB(K/M) ⊂ M(λs). 

When s satisfies (1) we have then Pλs,K/MB(K/M) = M(λs).


4. The second-order Hua operator 

We shall define the Hua operator both in terms of the enveloping algebra and using the covariant 

Cauchy–Riemann operator, the later having the advantage of being geometric and more 

explicit. To avoid some extra constants we fix and normalize the Killing form B on gC by requiring 

that on p + × p − it is given by 

B(u, ¯v) =〈u, v〉= 1 

p tr D(u, ¯v), u ∈ V = p+ , ¯v ∈ ¯V = p − , 

where the trace is computed on the space V . (So the standard Killing form, (X, Y ) ↦→ 

tr Ad(X) Ad(Y ), is−pB(X,Y).) 

Let {vj } and {v ∗ j } be dual bases of p+ and p − with respect to the normalized Killing form B. 

Let U(gC) be the enveloping algebra of gC. Since [p + , p − ]⊂kC, the operator 

H = HkC =−viv 

∗ j ⊗[vj ,v ∗ i ] 

i,j 

is an element of U(gC) ⊗ kC, and is independent of choice of the basis; it is called the secondorder 

Hua operator. If we identify U(gC) with left-invariant differential operators on G, H defines 

a homogeneous operator from C∞ (G/K) to the C∞-sections of G ×K kC. H can also be 

viewed as a differential operator from C∞ (G) to C∞ (G, kC). 

For X ∈ kC, define 

H X =− 

[X, vj ]v ∗ j ∈ U(gC). 

j 

Let S be a linear subspace of k, SC its complexification. Let {Xj } be a basis of SC and {X∗ j } 

be the dual basis with respect to the Killing form B. Then the projection of H onto U(gC) ⊗ SC 

is 

 

= H Xj ∗ 

⊗ Xj . 

HSC 

It can also be defined independently of basis, see e.g. [8, Proposition 1]. 

Symbolically we may write 

where 

j 

H = D b(z, ¯z)¯∂,∂ , 

b(z, ¯w) = 1 − D(z, ¯w) + Q(z)Q( ¯w) (12) 

is the Bergman operator, see [3]. (Operator H can also be defined by using the covariant Cauchy– 

Riemannian operator b(z, ¯z)¯∂, see [2,14,20]. For brevity we will not go into the details.) Using


an orthonormal basis {ej } of V with respect to the Hermitian scalar product (4), operator H can 

also be written as 

Hf(z)= 

D 

b(z, ¯z)ēi,ej ¯∂i∂j f(z). 

5. The Poisson kernel and the second-order Hua operator 

i,j 

We will compute the action of the Hua operator on the Poisson kernel. Let us first recall the 

notion of quasi-inverse in the Jordan triple V ; see [9]. Let z ∈ V and ¯w ∈ ¯V . The element z is 

called quasi-invertible with respect to ¯w, ifb(z, ¯w) is invertible. The quasi-inverse of z with 

respect to ¯w is then given by 

z ¯w = b(z, ¯w) −1 z − Q(z) ¯w . 

For example, in the type Ir,r+b case (see Section 2.2), let x,y ∈ V = Mr,r+b(C), then 

b(x, ¯y)z = (I − xy ∗ )z(I − y ∗ x). 

If I − xy ∗ is invertible, then the quasi-inverse of x is 

x ¯y = b(x, ¯y) −1 x − Q(x) ¯y = (I − xy ∗ ) −1 (x − xy ∗ x)(I − y ∗ x) −1 = (I − xy ∗ ) −1 x. 

Fix a Jordan frame {cj }1jr and choose an orthonormal basis {eα} of V consisting of the 

frame {cj }1jr, orthonormal basis of each of the subspaces Vjk and an orthonormal basis of 

each of the subspaces Vj0. The following lemma can be easily proved by direct computations. 

Lemma 5.1. 

(1) For any irreducible bounded symmetric domain Ω it holds: 

(a) r α=1 D(eα, ēα) = pZ0. 

(b) 

eα∈Vjk D(eα, ēα) = a 2 [D(cj , ¯cj ) + D(ck, ¯ck)]. 

(2) If Ω is of type Ir,r+b, then 

eα∈Vj0 D(eα, ēα) (1) = bD(cj , ¯cj ) (1) . 

We need also the following lemma. 

Lemma 5.2. Let ¯w ∈ ¯V . For any complex number s, the holomorphic and the anti-holomorphic 

differential of the function z ↦→ h(z, ¯w) are given by 

∂h(z, ¯w) s =−sh(z, ¯w) s ¯w z , 

Proof. This is a consequence of the formula 

see [21, Proposition 3.1]. ✷ 

¯∂h(z, ¯w) s =−sh(z, ¯w) s w ¯z . 

¯w z =−∂ log det b(z, ¯w) 1/p =−∂ log h(z, ¯w),


Theorem 5.3. For u fixed in S, function 

satisfies the following differential equation: 

z ↦→ Ps,u(z) := P(z,u) s 

 

n 

HPs,u(z) = 

r s 

2 D b(z, ¯z) z ¯z − u ¯z , ¯z z −ū z 

n 

− 

r sp 

 

Z0 Ps,u(z). 

Proof. Choose a basis {eα} as in Lemma 5.1. Then 

HPs,u(z) = 

D 

b(z, ¯z)eα, ēβ ¯∂α∂βPs,u(z). 

According to Lemma 5.2, 

α,β 

 

h(z, z) 

∂Ps,u(z) = ∂ 

|h(z, u)| 2 

n 

r s 

=− n 

r s 

 

h(z, z) 

|h(z, u)| 2 

n 

r s¯z z z 

−ū , 

where we have identified (p− ) ′ with p + by the Hermitian form (4). Performing one more time 

differentiation, we get 

 

¯∂∂Ps,u(z) 

n 

= 

r s 

2 Ps,u(z) z ¯z − u ¯z ⊗ ¯z z −ū z − n 

r sPs,u(z)¯∂ ¯z z −ū z . 

Moreover, 

¯∂ ¯z z −ū z = ¯∂ ¯z z = ¯∂∂ log h(z, ¯z) −1 = b(z, ¯z) −1 Id, 

where Id is the identity form in (p + ) ′ ⊗ (p− ) ′ . Hence, 

 

¯∂∂Ps,u(z) 

n 

= 

r s 

2 Ps,u(z) z ¯z − u ¯z ⊗ ¯z z −ū z 

n 

− 

r s 

 

b(z, ¯z) −1 Id. 

Consequently 

 

n 

HPs,u(z) = 

r s 

2 

 

n 

− 

r s 

α,β 

 

α 

z ¯z − u ¯z ,eα 

 

D(eα, ēα) Ps,u(z) 

 

z z 

¯z −ū , ēβ D b(z, ¯z)eα, ēβ 

 

n 

= 

r s 

2 D b(z, ¯z) z ¯z − u ¯z , ¯z z −ū z 

n 

− 

r s 

 

pZ0 Ps,u(z), 

since 

α D(eα, ēα) = pZ0. ✷ 

If Ω is of tube type, then the genus p is given by p = 2 n r and Theorem 5.3 becomes:


Corollary 5.4. Let Ω be a tube type domain. For any u ∈ S, the function z ↦→ Ps,u(z) satisfies 

the Hua equation 

where I is the identity operator. 

2 n 

HPs,u(z) = 2 s(s − 1)Ps,u(z)I, (13) 

r 

This corollary has been proved also by Faraut and Korányi [3, Theorem XIII.4.4]. Notice that 

the first factor 2 in (13) is because in this case our Hua operator is twice the Hua operator of 

Faraut and Korányi. In fact, we are using the definition [u, ¯v]=D(u, ¯v) so that for tube domain 

it is twice the “square” operator ✷ of Faraut and Korányi. 

In [12, Theorem 4.1] Shimeno gives the following characterization of the image of Poisson 

transform for tube type domains. 

Theorem 5.5. Let Ω be a tube type domain. Suppose s ∈ C satisfies the following condition: 

 

−4 1 + j a 

2 

 

n 

+ (s − 1) /∈{1, 2, 3,...} for j = 0 and 1. 

r 

A smooth function f on Ω is the Poisson transform Ps of a hyperfunction on S if and only if f 

satisfies the following Hua equation 

2 n 

Hf = 2 s(s − 1)f Z0. 

r 

This is a slightly different formulation of Shimeno’s result. In fact, if s ′ denotes the Shimeno’s 

parameter, then our parameter s is 

s = r 

 

s 

2n 

′ + n 

 

. 

r 

6. The main result for type Ir,r+b domains 

In this section we restrict ourself to the case Ω = Ir,r+b. Recall that in Section 2.2 we have 

fixed a decomposition kC = k (1) 

C ⊕ k(2) 

C .WeletH(1) be the first component of the Hua operator H. 

Symbolically H (1) is given by 

and can be identified with the operator 

H (1) = D b(z, ¯z)¯∂,∂ (1) , 

(Ir − zz ∗ )¯∂z · (Ir+b − z ∗ z) · t ∂z 

introduced by Hua [5], since in this case b(z, ¯z)v = (I − zz ∗ )v(I − z ∗ z). 

We state now the main theorem of this section.


Theorem 6.1. Suppose s ∈ C satisfies the following condition: 

−4 b + 1 + j + (r + b)(s − 1) /∈{1, 2, 3,...} for j = 0 and 1. (14) 

A smooth function f on Ir,r+b is the Poisson transform Ps(ϕ) of a hyperfunction ϕ on S if and 

only if f satisfies the following Hua equation: 

where Ir is the identity matrix of rank r. 

H (1) f = (r + b) 2 s(s − 1)f Ir, (15) 

Note here that the constant r + b = n/r for the domain Ir,r+b. 

6.1. The necessity of the Hua equation (15) 

To show the necessity of the Hua equation it is sufficient to show that the function Ps,u satisfies 

(15) for every u ∈ S. 

Proposition 6.2. If Ω is of type Ir,r+b, then 

H (1) Ps,u(z) = (r + b) 2 s(s − 1)Ps,u(z)Ir. 

Proof. It is sufficient to prove the formula at z = 0. Specifying the result of Theorem 5.3 to the 

type Ir,r+b domain we get for any u ∈ S, 

H (1) 

n 

Ps,u(0) = 

r s 

2 D(u, u) (1) 

n 

− 

r sp 

 

Z (1) 

 

0 Ps,u(0). 

Now, obviously D(u, u) (1) = Ir and Z (1) 

0 

= r+b 

2r+b Ir. Therefore, 

H (1) Ps,u(0) = (r + b) 2 s(s − 1)Ps,u(0)Ir. ✷ 

6.2. The Hua operator and the eigenfunctions of invariant differential operators 

We give first the expression for the radial part of the Hua operator H (1) , i.e. its restriction to 

K-invariant functions. We fix a Jordan frame {cj } r j=1 , then every element of V can be written as 

z = k 

r 

j=1 

tj cj , 

with k ∈ K, and tj 0. If f is a function on Ω invariant under K, we write 

f(z)= F(t1,...,tr). 

The function F is a symmetric function of the variables t1,...,tr, defined on the unit cube 

0 tj < 1.


Proposition 6.3. Let Ω be the type Ir,r+b domain. Let f be C 2 and K-invariant function, then 

for a = r j=1 tj cj , 

H (1) f(a)= 

r 

Hj F(t1,...,tr)D(cj , ¯cj ) (1) , (16) 

j=1 

where the scalar-valued operators Hj are given by 

Hj = 1 − t 2 

2 ∂2 j 

∂t2 j 

k=j 

+ 1 

tj 

∂ 

∂tj 

+ 

2 2 

1 − tj 1 − tk 

+ 2b 1 − t 2 1 

j 

tj 

∂ 

. 

∂tj 

 

1 

tj − tk 

∂ 

∂tj 

− ∂ 

 

+ 

∂tk 

1 

 

∂ 

+ 

tj + tk ∂tj 

∂ 

 

∂tk 

Proof. The proof is similar to the proof of [3, Theorem XIII.4.7] and we will only show how 

one can compute the last term of the radial part H (1) 

j , namely 2b(1 − t 2 j ) 1 tj 

∂ 

∂tj .Let{eα} be an 

orthonormal basis of V consisting of the frame {cj } r j=1 , an orthonormal basis of each of the 

subspaces Vjk and an orthonormal basis of each of the subspaces Vj0; and let zα = xα + iyα be 

the complex coordinates. Let f beafunctiononΩ and fix a = r k=1 tkck. Then 

H (1) f(a)= 

α,β 

For any X ∈ g and any v ∈ V , it is known that 

D (1) ∂ 

b(a,ā)eα, ēβ 

2 

∂zα∂ ¯zβ 

∂Xv∂Xvf + ∂ X 2 v f = 0. 

f(a). 

We will apply this formula for different elements in g. Suppose eα = eβ ∈ Vj,0. For the element 

X = i(D(eα, ¯cj ) + D(cj , ēα)) ∈ k,wehave 

and 

Therefore 

which implies 

Xa = i D(eα, ¯cj ) + D(cj , ēα) a = iD(eα, ¯cj )a = iD(a,ēα)eα = itj eα, 

X 2 

a = X(itjeα) =−tj D(eα, ¯cj ) + D(cj , ēα) a =−tjcj . 

∂itjeα ∂itjeα f(a)+ ∂−tj cj f(a)= 0, 

∂2 ∂y2 α 

f = 1 ∂ 

F. 

tj ∂tj


Similarly, for X = D(eα, ¯cj ) − D(cj , ēα) ∈ k,wehave 

and 

Hence, 

From this we obtain 

Summarizing, we find on Vj,0, 

Xa = D(eα, ¯cj ) − D(cj , ēα) a = tj eα, 

X 2 

a = X(tjeα) = tj D(eα, ¯cj ) − D(cj , ēα) eα =−tjcj . 

∂ 2 

4 f = 


∂tj eα∂tj eαf(a)+ ∂−tj cj f(a)= 0. 

∂2 ∂x2 α 

f = 1 ∂ 

F. 

tj ∂tj 

0 if α = β, 

∂ 2 

∂x 2 α 

+ ∂2 

∂y2 

1 ∂ 

f = 2 F if α = β. 

α 

tj ∂tj 

Furthermore, 

D (1) 2 (1) 2 

b(a,ā)eα, ēα = D 1 − tj eα, ēα = 1 − tj D(eα, ēα) (1) . 

Hence, 

r 

 

j=1 eα,eβ∈Vj,0 

= 

= 2 

r 

 

j=1 eα∈Vj,0 

= 2b 

D (1) ∂ 

b(a,ā)eα, ēβ 

2 


2 

1 − tj D(eα, ēα) (1) 2 1 

r 2 1 ∂F 

1 − tj tj ∂tj 

j=1 

 

eα∈Vj,0 

tj 

f(a) 

∂F 

∂tj 

D(eα, ēα) (1) 

r 2 1 ∂F 

1 − tj D(cj , ¯cj ) 

tj ∂tj 

(1) , 

j=1 

since we already proved in Lemma 5.1, that 

 

D(eα, ēα) (1) = bD(cj ,cj ) (1) . 

This finishes the proof. ✷ 

eα∈Vj,0


The next proposition claims that the Hua equation (15) for s ∈ C is sufficient for f being an 

eigenfunction of D(Ω) G . A similar result for general tube domains is proved in [12]. 

Proposition 6.4. Let Ω be the type Ir,r+b domain. Let s ∈ C and let λs be given by (11). Suppose 

f on Ω satisfies the Hua equation (15). Then f is an eigenfunction of all T ∈ D(Ω) G with 

eigenvalues χλs (T ). 

Proof. Let f be a function on Ω solution of the Hua equation. Let g ∈ G, then the function 

 

Φ(z) = f(gk· z) dk, z ∈ Ω, 

is a K-biinvariant solution of differential system, 

K 

Hj Φ = (r + b)s(s − 1)Φ, j = 1,...,r. 

Thus by a result of Yan [18], 1 Φ is proportional to the unique spherical function 

 

ϕλs (z) = 

K 

eλs (k · z) dk 

in M(λs), i.e.Φ(z) = cϕλs (z). It is easy to see that c = f(g· 0), then 

 

f(gk· z) dk = ϕλs (z)f (g · 0); 

K 

and consequently, by [4, Chapter IV, Proposition 2.4], f is a joint eigenfunction of all T ∈ 

D(Ω) G with eigenvalues χλs (T ). ✷ 

6.3. The sufficiency of the Hua equation (15) 

We suppose in the rest of Section 6 that s ∈ C satisfies condition (14) and that f satisfies 

the sufficient condition (15) in Theorem 6.1. It follows immediately from Proposition 6.4 that 

f ∈ M(λs), and thus by Kashiwara et al. [7], f is the Poisson transform of a function ϕ on the 

Furstenberg boundary G/Pmin, f = Ps(ϕ). To prove that ϕ is a function on the Shilov boundary 

S, we follow a method by Berline and Vergne [1] (see also [8]), the reader is refereed that 

paper for some general arguments. 

We need first two elementary lemmas for general bounded symmetric domain Ω; the first 

one gives explicit formulas for the root spaces g α , α ∈ Σ(g, a), and can easily be deduced from 

the Peirce decomposition (see [9,15,16]). The second is essentially stated in [1] in terms of the 

Cayley transform, it has however an easier form in terms of the Jordan triple and can easily be 

proved using the first. To state them we need some notational preparation. Recall the quadratic 

1 Roughly speaking, the differential equations used in [18] is obtained from (16) by the change of coordinates xj = 

−t2 j /1 − t2 j .


map z → Q(z) given in Section 2. For the fixed Jordan frame {cj } and the corresponding Peirce 

decomposition (5), the map 

τ : z → τ(z)= Q(e)¯z, 

where e = c1 +···+cr, defines a real involution of V2 and thus a real form 

A(e) = z ∈ V : τ(z)= z 

of V2; letV2 = A(e) ⊕ iA(e) be corresponding decomposition with A(e) being a real Jordan 

algebra. Let 

then A(e) = iB(e).For1 j k r, let 

B(e) = z ∈ V : τ(z)=−z , 

Vjk = Ajk ⊕ Bjk 

be the decomposition of the space Vjk into real and imaginary part relative to the real form A(e). 

Lemma 6.5. The root spaces g α , α ∈ Σ(g, a), are explicitly given as follows: 

for 1 j,k r. 

g ±βj 

 

= R ξicj ∓ 2iD(cj , ¯cj ) , 

g (βk−βj 

)/2 

= ξa + D(ck − cj , ā): a ∈ Ajk , 

g ±(βk+βj 

)/2 

= ξb ∓ D(ck + cj , ¯b): b ∈ Bjk , 

g ±βj 

 

/2 

= ξv ± (D(cj , ¯v) − D(v, ¯cj ): v ∈ V0j 

Lemma 6.6. The corresponding root spaces for the positive compact roots γk−γj 

2 , 1 j


therefore, system (15) implies in particular 

However, 

Lemma 6.7. We have 

H (1) 

k + 

C 

= 

k>j 

H (1) 

k + 

C 

H (1) 

k (γ k −γ j )/2 

C 

f = 0. (17) 

+ 

γj /2 

(1) 

kC = 0. 

r 

H 

j=1 

(1) 

k γj /2 

C 

Proof. Indeed, using Lemma 6.6, let D(cj , ¯v) ∈ k γj /2 

C , with v = ej,j+m ∈ Vj,0 (m>0). Then 

Hence, from (17) it follows 

Lemma 6.8. We have 

D(cj , ¯v) (1) = ej,je ∗ j,j+m = 0. ✷ 

k + 

C 

 

H 

k>j 

(1) 

k (γk−γj )/2 

C 

(1) = 

k>j 

. 

f = 0. (18) 

k (γk−γj )/2 

C 

and the right-hand side is a linear direct sum, namely the spaces (k (γk−γj )/2 

C ) (1) are linearly 

independent. 

Proof. This follows easily from Lemma 6.6 by observing that D(ck, ¯v) ↦→ D(ck, ¯v) (1) is a linear 

homomorphism, and that D(ck, ¯v) (1) =¯αek,j for v = αej,k + bek,j ∈ Vj,k. ✷ 

We conclude from (18) and the above lemma that 

for any positive compact root (γk − γj )/2. 

Let Ψ +,(i) 

c 

H (1) 

k (γk−γj )/2 

C 

(1), 

f = 0 (19) 

be the set positive compact roots in k (i) 

C ,fori = 1, 2. Then (19) implies 

H β f = 0 forβ ∈ Ψ +,(1) 

c 

with β ≡ γk − γj 

2 

(k > j),


where H β is the component of H given by 

H β = 

[Eβ,vα]¯vα 

α∈Ψ + n 

and Eβ is the root vector of β. 

Now we fix for the rest of this section β ∈ Ψ +,(1) 

c such that 

β| t − 

2 

C = γj − γj−1 

The root vector Eβ has the form Eβ = D(cj , ¯w) with w = ej,j−1 or w = ej−1,j being one of the 

basis vectors {vα}. Observe that [Eβ,vα]=0 unless α is in the set Ψ1 ∪ Ψ2 ∪ Ψ3 where 

 

Ψ1 = α ∈ Ψ + n : α| t − 

 

Ψ2 = α ∈ Ψ + n : α| t − 

2 

C = γk + γj−1 

2 

C = γk + γj−1 

 

Ψ3 = α ∈ Ψ + n : α| t − 

 

γj−1 

= . 

C 2 

Consider the Poincaré–Birkhoff–Witt decomposition 

and let π be the projection 

. 

 

,k j − 1 , 

 

,k j , 

U(gC) = U(gC)kC + U(aC + n − 

C ) 

π : U(gC) = U(gC)kC + U(aC + n − 

C ) → U(aC + n − 

C ). 

The function f is now viewed as a function on G = NAK, and the group A will be identified 

as (R + ) r . Under this identification, f satisfies, furthermore, the equation 

R π H β f = 0, 

where R is the mapping from U(aC + nC) to differential operators on NA defined by 

∂ 

R(ξk) = tk , R(X−α) = t 

∂tk 

α X−α, 

for ξk ∈ a, 1 k r, and X−α ∈ n identified with the corresponding left-invariant differential 

operator. 

We will prove that operator t − 1 2 (βj −βj−1) R(π(H β )) has analytic coefficient near t = 0 and 

study the induced equation of t − 1 2 (βj −βj−1) R(π(H β ))f = 0.


6.3.1. The projection of the Hua operator in the PBW-decomposition 

We will compute the Poincaré–Birkhoff–Witt components of the Hua operators as an element 

in the universal algebra of gC. Let now Ω be a general bounded symmetric domain. 

Lemma 6.9. The Iwasawa decomposition of v ∈ p + and ¯v ∈ p − in gC = aC + n − 

C + kC is given 

as follows. 

(1) For v ∈ Vkj , r k>j 1, 

v = ζv + ζ ′ v − D(v, ¯ck), ¯v = η ¯v + η ′ ¯v − D(ck, ¯v), 

where ζv,η¯v ∈ g −(βk−βj )/2 

C , ζ ′ v ,η′ ¯v ∈ g−(βk+βj )/2 

C are given by 

(2) For v = cj ∈ Vjj , 1 j r, 

with 

(3) For v ∈ Vj0, 1 j r, 

with 

ζv = 1 

v − τ(v)+ D(v, ¯ck −¯cj ) 

2 

, 

ζ ′ 1 

v = v + τ(v)+ D(v, ¯cj +¯ck) 

2 

, 

η ¯v = 1 

¯v − τ(v)+ D(cj − ck, ¯v) 

2 

, 

η ′ 1 

¯v = τ(v)+¯v + D(cj + ck, ¯v) 

2 

. 

cj = 1 

2 ξj − 1 

2 ζj − D(cj , ¯cj ), ¯cj =− 1 

2 ξj − i 

2 ζj − D(cj , ¯cj ), 

ζj = i ξicj + 2D(cj , ¯cj ) . 

v = ζv − D(v, ¯cj ), ¯v = η ¯v − D(cj , ¯v), 

ζv = v + D(v, ¯cj ) ∈ n −βj /2 

C , η¯v =¯v + D(cj , ¯v) ∈ n −βj /2 

C 

We denote by π n 0 C the projection onto the nilpotent subalgebra 

n 0 C 

= 

k>j1 

g −(βk−βj )/2 

C 

.


in the Iwasawa decomposition of gC, 

Then, it follows from Lemma 6.9, 

gC = kC + aC + 

kj0 

πn0 ( ¯v) =−π 

C 

n0 C 

g −(βk+βj )/2 

C + n 0 C . 

τ(v) , (20) 

which we will need in the next proposition. 

Return back to type Ir,r+b domains. We compute now the projection π(H β ) of H β . Recall 

that the β-root vector is Eβ = D(cj , ¯w), with w = ej,j−1 or w = ej−1,j . 

Proposition 6.10. The projection π(H β ) is given by 

where the last term 

π H β j−2 

= 

 

k=1 vα∈Vk,j−1 

(ζ{cj ¯wvα} + ζ ′ {cj ¯wvα} )(η ¯vα + η′ ¯vα ) + jη¯w + jη ′ ¯w 

+ (ζτ(w) + ζ ′ τ(w) ) 

 

− 1 

2 ξj−1 − i 

2 ζj−1 

 

+ 

r 

(ζ{cj ¯wvα} + ζ ′ {cj ¯wvα} )(η ¯vα + η′ ¯vα ) 

+ 

k=j+1 vα∈Vk,j−1 

 

1 

2 ξj − 1 

2 ζj 

 

(η ¯w + η ′ 

¯w ) + 

vα∈Vj−1,0 

 

b(η ¯w + η 

Jβ = 

′ ¯w ) if w = ej−1,j , 

0 if w = ej,j−1, 

ζ{cj ¯wvα}η ¯vα 

and where the sum 

vα∈Vkj is taken over the orthonormal basis {vα}of Vk,j. 

Proof. We compute the projection 

α∈Ψ + n π([Eβ,vα]v∗ α ). For α ∈ Ψ + n 

[Eβ,vα]=0 unless α ∈ Ψ1 ∪ Ψ2 ∪ Ψ3. 

Case I. α ∈ Ψ1, with α| − 

tC = (γk + γj−1)/2. Then vα ∈ Vk,j−1 and 

vβ+α := [Eβ,vα]= 

D(ci, ¯w),vα = D(ci, ¯w)vα ∈ Vj,k. 

By the previous lemma, for k


[Eβ,vα]v ∗ α = vβ+α ¯vα ≡ (ζvβ+α + ζ ′ vβ+α )(η ¯vα + η′ ¯vα ) − 

D(vβ+α,cj ), ¯vα 

= (ζvβ+α + ζ ′ vβ+α )(η ¯vα + η′ ¯vα ) + D(cj , ¯vβ+α)vα. 

To find the last term we note first that for any Jordan triple system [9], 

 

D(vα, ¯vα), D(w, ¯cj ) = D 

vα, D(cj , ¯w)vα − D D(w, ¯cj )vα, ¯vα ; 

we let it act on cj and then sum over vα 

 

vα∈Vk,j−1 

D cj ,D(cj , ¯w)vα 

 

vα = a 

D(cj−1,cj−1) + D(ck,ck) 

2 

w 

by using Proposition 5.1. It is further w, since a = 2 for type I domains. Thus 

 

≡ 

 

(ζvβ+α + ζ ′ vβ+α )(η ¯vα + η′ ¯vα ) + (j − 2)η ¯w + (j − 2)η ′ ¯w . 

vα∈Vk,j−1 

k


Now let k = j, then [Eβ,vα]=D(cj , ¯w)vα = D(vα, ¯w)cj =〈vα,w〉cj , which vanishes except 

when vα = w and in that case, 

and 

 

1 

[Eβ,vα]¯vα = cj ¯vα = 

2 ξj − 1 

2 ζj 

 

− D(cj , ¯cj ) ¯w, 

 

1 

[Eβ,vα]¯vα ≡ 

2 ξj − 1 

2 ζj 

 

(η ¯w + η ′ ¯w ) + η ¯w + η ′ ¯w . 

1 

Case III. α ∈ Ψ3, with α| − 

t = 

C 2γj−1, and the root vector vα ∈ Vj−1,0. In this case, we have 

[Eβ,vα]¯vα = D(cj , ¯w)vα ¯vα ≡ ζD(cj , ¯w)vα − D 

D(cj , ¯w)vα,cj η ¯vα 

≡ ζD(cj , ¯w)vαη ¯vα + D 

cj , D(cj ,w)vα vα. 

However, by the commutator relation (JP15) in [9] we have 

 

D(w, ¯vα), D(cj , ¯cj ) = D 

D(w, ¯vα)cj , ¯cj − D cj , D(vα, ¯w)cj =−D cj , D(vα, ¯w)cj 

since D(w, ¯vα)cj = 0 by the Peirce rule that D(w, ¯vα)cj ∈{Vj,j−1 ¯Vj−1,0Vjj }={0}, thus 

D 

cj , D(cj , ¯w)vα vα = D(cj ,cj ), D(w, ¯vα) vα = D(cj ,cj )D(w, ¯vα)vα = D(vα, ¯vα)w 

since D(cj ,cj )vα = 0. 

It is easy to see, by direct matrix computation, that, 

Hence, modulo U(gC)kC 

Consequently, 

 

 


 


[Eβ,vα]v 


∗ α 

and this finishes the proof. ✷ 

 

b ¯w if w = ej−1,j , 

D(vα, ¯vα)w = 

0 ifw = ej,j−1. 

 

b(η ¯w + η 

D(vα, ¯vα)w ≡ 

′ ¯w ) if w = ej−1,j , 

0 if w = ej,j−1. 

≡ 

vα 

ζD(cj , ¯w)vαη ¯vα + 

 

b(η ¯w + η ′ ¯w ) if w = ej−1,j , 

0 if w = ej,j−1


6.3.2. The induced equation 

We apply now the theory of boundary values of eigenfunctions of D(Ω) G on symmetric 

spaces, see [7,10,11,13]. 

We identify the space G/K with NA and A with (R + ) r . It follows from Proposition 6.10 that 

operator t − 1 2 (βj −βj−1) R[π(H β )] has analytic coefficients near t = 0. Then the induced equation 

for the differential equation t − 1 2 (βj −βj−1) R[π(H β )]f = 0is 

where t = (t1,t2,...,tr) ∈ A = (R + ) r , and 

lim 

t→0 tλ−ρ t − 1 2 (βj 

−βj−1) β 

R π H t ρ−λ (Bλf)= 0, 

t μ = t μ(ξ1) 

1 ···t μ(ξr ) 

r 

for μ ∈ a ∗ C .HereBλf is the boundary value of f . 

Proposition 6.11. The boundary value Bλf of f satisfies the following induced equation: 

R[ζτ(w)](Bλf)= 0. (21) 

Observe, using (20), that the induced Eq. (21) is equivalent to the following one: 

R[η ¯w](Bλf)= 0. 

Proof. Let us compute the limit of the differential operator 

t λ−ρ t − 1 2 (βj −βj−1) R π H β t ρ−λ 

when t → 0. We will consider each term in the projection π(H β ). 

• The differential operator corresponding to j(η¯w + η ′ ¯w ) is 

t λ−ρ t − 1 2 (βj −βj−1) j t 1 2 (βj −βj−1) R(η ¯w) + t 1 2 (βj +βj−1) R(η ′ ¯w ) t ρ−λ , 

and its limit when t ↦→ 0is 

jR(η ¯w). (22) 

• Consider the quadratic term (ζτw + ζ ′ 1 

τw )(− 2ξj−1 − i 

2ζj−1). The corresponding differential 

operator is 

t λ−ρ t − 1 2 (βj 

 

t 1 

−βj−1) 

2 (βj −βj−1) 

R(ζτ(w) + t 1 2 (βj +βj−1) ′ 

R(η τ(w) ) 

 

× − 1 

2 tj−1 

∂ 

∂tj−1 

− i 

2 tβj−1 

R(ζj−1) t ρ−λ 

× R(ζτ(w)) + t βj−1 R(ζ ′ τ(w) ) 

− 1 

2 (ρ − λ)(ξj−1) − i 

2 tβj−1 R(ζj−1) 

 

.

Its limit when t → 0is 


− 1 

(ρ − λ)(ξj−1)R(ζτ(w)). 

2 

• For the quadratic term ( 1 2ξj − 1 2ζj )(η ¯w − η ′ ¯w ), the corresponding differential operator is 

t λ−ρ t − βj −β 

j−1 1 

2 

2 tj 

Its limit is 

which is 

lim 

t→0 tλ−ρ t − βj −βj−1 2 

= lim 

∂ 

∂tj 

t 

t→0 λ−ρ t − βj −βj−1 2 

− 1 

2 tβj 

t βj −βj−1 R(ζj ) 2 R(η ¯w) + t βj +βj−1 2 R(η ′ ¯w )t 

ρ−λ . 

 

1 ∂ βj −βj−1 t 2 t 

2 ∂tj 

ρ−λ R(η ¯w) + t βj +βj−1 2 t ρ−λ R(η ′ ¯w ) 

 

1 βj − βj−1 

(ξj ) + (ρ − λ)(ξj ) t 

2 2 

βj −βj−1 2 t ρ−λ R(η ¯w) 

+ 1 

 

βj + βj−1 

(ξj ) + (ρ − λ)(ξj ) t 

2 2 

βj +βj−1 2 t ρ−λ R(η ′ ¯w ) 

 

, 

1 

1 + (ρ − λ)(ξj ) 

2 

R(η ¯w). 

• The induced equation corresponding to the last term of the projection π(Hβ ) is 

 

bR(η ¯w), 

(23) 

0. 

• Now, it is easy to see, using the same computations, that the induced equation of the remaining 

terms of π(Hβ ) is zero. 

It follows now from (22), (23) and (20), that the boundary value Bλf of f satisfies 

where C1 is given by 

C1R(ζτ(w))(Bλf)= 0, 

C1 = 1 

2 (ρ − λ)(ξj 

 

12 

+ j + b, 

− ξj−1) + 

1 

2 + j. 

Now if C1 = 0, then the induced equation is R(ζτ(w))(Bλf)= 0. On the other hand, if C1 = 0, 

we may replace f by tκ(β j −βj−1 2 ) f for sufficiently large κ>0, consider the differential operator 

t − 1 2 (βj −βj−1) κ( 

t βj −βj−1 2 

) β 

R π H t −κ(β j −βj−1 2 ) 

, 

and we still prove that R(ζτ(w))(Bλf)= 0, see also [12]. ✷


We continue the proof of the necessity condition of the Hua equation. We now get 

R(ζτ(w))(Bλs f) = 0 for any root β such that β ≡ 1 2 (γj − γj−1). Since { 1 2 (γj − γj−1), 

2 j r} is the set of simple roots of the system { 1 2 (γk − γj ), 1 j


Let f be in M(λs) and suppose f satisfies (25). Then f = Ps(ϕ) is the Poisson transform of a 

hyperfunction ϕ on S. 

7.2. The necessity of the Hua equation (25) 

Consider the operator 

Proposition 7.3. Let s ∈ C. We have 

and 

In particular, for σ = 0,(p+ b)/2, 

Y = U − −2σ 2 + 2pσ + c 

σ(2σ − p − b) W. 

WPs,u(z) = Ps,u(z)σ 2 (2σ − p − b)u 

UPs,u(z) = Ps,u(z)σ −2σ 2 + 2pσ + c u. 

YPs,u(z) = 0, 

and the image f = Ps(ϕ) of the Poisson transform of a hyperfunction ϕ on S satisfies 

Yf = 0. 

Proof. We compute first W on Ps(z, u). By the covariant property of W and transformation 

property (10) of the kernel Ps(z, u) we need only to prove that the formula is valid at z = 0. We 

use the differentiation h(z, z)¯∂ in place of ¯∂ as it will produce some more compact formulas; the 

eventual result will be the same at z = 0. (The operator h(z, z)¯∂ can be geometrically defined, 

see Section 4.) Proceeding as in the proof of Theorem 5.3 by using Lemma 5.2, we have 

 

h(z, z)¯∂ ∂Ps(z, u) = σ 2 Ps,u(z) b(z, z) z ¯z − u ¯z ⊗ ¯z z −ū z − σPs,u(z)Id. 

Its image under −Ad V ⊗ ¯V →kC is, 

−Ad V ⊗ ¯V →kC 

since 

and 

h(z, z)¯∂ ∂Ps(z, u) = σ 2 Ps,u(z)D b(z, z) z ¯z − u ¯z , ¯z z −ū z − σPs,u(z)pZ0, 

−AdV ⊗ ¯V →kC 

(u ⊗¯v) =−[u, ¯v]=D(u, ¯v) 

−AdV ⊗ ¯V →kC 

Id = pZ0. 

To compute ¯∂Ad V ⊗ ¯V →kC (h(z, z)¯∂)∂Ps(z, u) at z = 0 we observe that for any function f , 

¯∂f(0) is the coefficient of ¯z in the expansion of f near z = 0. By direct computation we find:


−¯∂AdV ⊗ ¯V 

¯∂∂Ps(z, →kC 

u)|z=0 = σ 3 u ⊗ D(u,ū) − pσ 2 u ⊗ Z0 

+ σ 

2 

j 

vj ⊗ D Q(u) ¯vj , ū − D(u, ¯vj ) , 

where {vj } is an orthonormal basis of V .Nowforv ∈ V,X∈ kC, AdV ⊗kC→(v ⊗ X) =[v,X]= 

−Xv, where Xv is the defining action of kC on V . We get 

AdV ⊗kC→V ¯∂AdV ⊗ ¯V 

¯∂∂Ps(z, →kC 

u)|z=0 

= 2σ 3 u − pσ 2 

2 

u + σ D Q(u) ¯vj , ū 

vj − D(u, ¯vj )vj . 

j 

The sum can further be evaluated by using the Peirce decomposition with respect to u, and we 

obtain eventually 

AdV ⊗kC→V ¯∂AdV ⊗ ¯V 

¯∂∂Ps(z, →kC 

u)|z=0 = σ 2 (2σ − p − b)u. 

This proves the first formula. 

For the second formula we have first 

and 

¯∂Ps(z, u) = σPs(z, u)b(z, z) z ¯z − u ¯z 

∂ ¯∂ ¯∂Ps(z, u)(0)|z=0 = σ 

j,k 

Performing the differentiation we find then 

So that 

∂vk ¯∂vj 

∂ ¯∂ ¯∂Ps(z, u)|z=0 = σ 3 ū ⊗ u ⊗ u − σ 

Ps(z, u)b(z, z) z ¯z − u ¯z |z=0. 

2 

− σ 

¯vk ⊗ vj ⊗ D(vk, ¯vj )u. 

j,k 

UPs(z, u)|z=0 =−σ 3 D(u,ū)u + σ 

k 

2 

+ σ 

D(vj , ¯vk)D(vk, ¯vj )u. 

Again 

k D(vk, ¯vk)v = pv for v ∈ V and 

 

D(vj , ¯vk)D(vk, ¯vj )u = cp 

by [19, Lemma 2.5] with c given as in (24). ✷ 

j,k 

j,k 

k 

¯vk ⊗ (vk ⊗ u + u ⊗ vk) 

 

D(vk, ¯vk)u + D(u, ¯vk)vk


In particular, if s = 1, WPs,u(z) = 0; the similar result, VPs,u(z) = 0 for the Hua operator V 

was proved by Berline and Vergne [1, Proposition 3.3]. 

7.3. The sufficiency of the Hua equation (25) 

The idea of the proof is similar to that in Section 6, and many technical computations on the 

various decomposition involving the third-order Hua operators W and U are parallel to those 

in [1] for the Berline–Vergne’s Hua operator V, so we will not present all details. 

Suppose hereafter that f ∈ M(λs) satisfies (25). We first observe that the operator U can also 

be written as 

U = 

vγ v ∗ αv∗ β ⊗ vα, [vβ,v ∗ γ ] 

α,β,γ 

since [[v ∗ γ ,vα],vβ]=[vα, [vβ,v ∗ γ ]] by the Jacobi identity and by [vα,vβ]=0. Writing 

with 

U δ ⊗ vδ = 

α+β−γ =δ 

we have, modulo U(gC)kC, 

U = U δ ⊗ vδ, W = W δ ⊗ vδ, 

vγ v ∗ α v∗ β ⊗ vδ, W δ ⊗ vδ = 

U δ ⊗ vδ − W δ 

 

⊗ vδ = 



|Cα,β,γ | 2 

 

v ∗ δ ⊗ vδ, 

v ∗ α v∗ β vγ ⊗ vδ, (26) 

where Cα,β,γ are given by [vα, [vβ,v ∗ γ ]] = Cα,β,γ vδ. 

Writing Y = 

δ Yδ ⊗ vδ as above with Y δ ∈ U(g) we have then modulo U(gC)kC 

with 

Thus 

C1 := 


Y δ = C1v ∗ δ 

δ + C2W 

|Cα,β,γ | 2 , C2 := 1 − −2σ 2 + 2pσ + c 

σ(2σ − p − b) . 

Y δ f = 0. (27) 

for any non-compact root δ ≡ (γj + γj−1)/2 modulo t − 

C , by our assumption on f . We will henceforth 

fix one such δ, and study the induced equation of (27).


Recall projection π from U(gC) onto U(aC + n − 

C ). Here we will not be able to find explicit 

formula for π(Yδ ) as in Proposition 6.10. Nevertheless we can compute the induced equation. 

Consider the decomposition of U(aC + n − 

C ) under aC, 

where 

is root lattice of Σ − (g, a). 

U(aC + n − 

 

C ) = 

p∈Π − 

U(aC + n − 

C )p, (28) 

Π − 

= p = 

β∈Σ − (g,a) 

 

cββ, 0 cβ, cβ∈ Z 

Lemma 7.4. Let α + β − γ = δ ≡ (γj + γj−1)/2 be as in (26). Decomposing π(¯vα ¯vβvγ ) ∈ 

U(aC + n − 

C ) according to (28), 

π(¯vα ¯vβvγ ) = 

π(¯vα ¯vβvγ )p, (29) 

p∈Π − 

we have that p −(βj − βj−1)/2, for any p appearing in (29) so that π(¯vα ¯vβvγ )p = 0. 

Proof. The lemma can be proved by a case by case computation of the projection by using 

Lemma 6.9, and is essentially contained in [1]. We sketch another somewhat more systematic 

method. We denote the Iwasawa decomposition as ¯vα = π(¯vα) + y with y ∈ kC. Thus 

The Iwasawa projection of the first term is 

¯vα ¯vβvγ = π(¯vα) ¯vβvγ + y ¯vβvγ . 

π(¯vα)π( ¯vβvγ ). 

The projection of the second term is 

π 

[y, ¯vβ]vγ + π ¯vβ[y,vγ ] . 

Observe by Lemma 6.9 that the element y is a positive compact root vector, so that all these projections 

involved are of the form π(¯vδvɛ) with vδ and vɛ being non-compact positive root vectors. 

Our lemma reduces to the following claim, which can be proved easily by using Lemma 6.9. The 

weights p of π(¯vδvɛ) satisfy the inequalities 

p − βj − βj−1 

2 

+ βk − βk ′ 

2 

p − βj − βj−1 

2 

if δ − ɛ = γj + γj−1 

2 

if δ − ɛ = γj + γj−1 

2 

− γk + γk ′ 

2 

− γk, 

with k>k ′ ,

and 


p − βj − βj−1 

2 

+ βk 

2 

if δ − ɛ = γj + γj−1 

2 

− γk 

. ✷ 

2 

From this it follows that the operator t − 1 2 (βj −βj−1) R[π(Y δ )] has analytic coefficients in t near 

t = 0 and thus the induced equation 

lim 

t→0 tλ−ρ t 1 2 (βj −βj−1) R π Y δ t ρ−λ (Bλf)= 0 (30) 

of the equation Yδf = 0 is well defined. 

Consider next the eigenspace decomposition of the space n − 

rj=1 ξj : 

1 

2 

with 

n −1 

C 

= 

kj1 

and correspondingly 

Let 

n − 

C 

= n−1 

C 

g −(βk+βj )/2 

C , n −1/2 

C 

U(aC + n − 

C ) = U(aC + n − 

+ n−1/2 

C + n0 C 

 

= 

k1 

C )n −1 

C 

π Y δ = Y1 + Y0 

g −βk/2 

C , n 0 C 

C under the element 1 2 ξc = 

= 

kj 

g −(βk−βj )/2 

C ; 

 

+ n−1/2 

C + U aC + n 0 

C . (31) 

be the decomposition of π(Yδ ) according to (31). As the decompositions (28) and (31) are consistent, 

we see that the weights p that appear in the decomposition of Y1 according to (28) satisfy 

p −δ, and p μ for μ such that n μ 

C ⊂ n−1 

C + n−1/2 

C . The first implies that the induced equation 

is well defined, and the second that the induced Eq. (30) now reduces to 

lim 

t→0 tλ−ρ t − 1 2 (βj −βj−1) R[Y1]t ρ−λ (Bλf)= 0. (32) 

The element Y0 can be found along the same lines as in [1], where the constant term C1ζvδ 

was found. 

Lemma 7.5. The element Y0 is given by 

 

1 

2 

C1 + C2 −ξj − ξ 

2 

2 j−1 + ξj 

 

′ 

ξj−1 + C 1ξ ζvδ , 

where ξ ∈ aC, C ′ 1 is some constants independent of λ.


In particular the induced Eq. (32) is of the form 

 

C1 + C2D(λ) R(ζvδ )(Bλf)= 0, 

where 

D(λ) = 1 

−(ρ − λ)(ξj ) 

2 

2 − (ρ − λ)(ξj−1) 2 + (ρ − λ)(ξj )(ρ − λ)(ξj−1) 

+ C ′ 1 (ρ − λ)(ξ). 

Observe first that C1 > 0. If C2 = 0 it follows immediately that R(ζvδ )(Bλf)= 0, so we need 

only to consider the case C2 = 0. If C1 + C2D(λ) = 0 we get again R(ζvδ )(BλF) = 0. Finally, 

if C1 + C2D(λ) = 0 we may replace f by t κγj for sufficiently large κ and still prove 

that R(ζvδ )(Bλf)= 0; see [12]. This completes the proof. 

References 

[1] N. Berline, M. Vergne, Équations de Hua et noyau de Poisson, in: Noncommutative Harmonic Analysis and Lie 

Groups, Marseille, 1980, in: Lecture Notes in Math., vol. 880, Springer, Berlin, 1981, pp. 1–51. 

[2] M. Englis, J. Peetre, Covariant Laplacian operators on Kähler manifolds, J. Reine Angew. Math. 478 (1996) 17–56. 

[3] J. Faraut, A. Korányi, Analysis on Symmetric Cones, Oxford Math. Monogr., Clarendon Press, Oxford, 1994. 

[4] S. Helgason, Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical 

Functions, Pure Appl. Math., vol. 113, Academic Press, Orlando, FL, 1984. 

[5] L.K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Amer. Math. 

Soc., Providence, RI, 1963, iv+164 pp. 

[6] K. Johnson, A. Korányi, The Hua operators on bounded symmetric domains of tube type, Ann. of Math. (2) 111 

(1980) 589–608. 

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operators on a symmetric space, Ann. of Math. (2) 107 (1978) 1–39. 

[8] M. Lassalle, Les équations de Hua d’un domaine borné symétrique du type tube, Invent. Math. 77 (1984) 129–161. 

[9] O. Loos, Bounded symmetric domains and Jordan pairs, University of California, Irvine, 1977. 

[10] T. Oshima, A definition of boundary values of solutions of partial differential equations with regular singularities, 

Publ. Res. Inst. Math. Sci. 19 (1983) 1203–1230. 

[11] T. Oshima, Boundary value problems for systems of linear partial differential equations with regular singularities, 

in: Adv. Stud. Pure Math., vol. 4, 1984, pp. 391–432. 

[12] N. Shimeno, Boundary value problems for the Shilov boundary of a bounded symmetric domain of tube type, 

J. Funct. Anal. 140 (1996) 124–141. 

[13] N. Shimeno, Boundary value problems for various boundaries of Hermitian symmetric spaces, J. Funct. Anal. 170 

(2000) 265–285. 

[14] G. Shimura, Differential operators, holomorphic projection, and singular forms, Duke Math. J. 76 (1994) 141–173. 

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(1994) 563–597. 

[16] H. Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986) 1–25. 

[17] H. Upmeier, Jordan algebras in analysis, operator theory, and quantum mechanics, in: CBMS Reg. Conf. Ser. Math., 

vol. 67, Amer. Math. Soc., Providence, RI, 1987. 

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[19] G. Zhang, Invariant differential operators on Hermitian symmetric spaces and their eigenvalues, Israel J. Math. 119 

(2000) 157–185. 

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domains, J. Math. Kyoto Univ. 42 (2002) 207–221.



Symmetry of large solutions of nonlinear elliptic 

equations in a ball 

Alessio Porretta a,1 , Laurent Véron b,∗ 

a Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy 

b Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Université François Rabelais, 

Tours 37200, France 




Abstract 

Let g be a locally Lipschitz continuous real-valued function which satisfies the Keller–Osserman condition 

and is convex at infinity, then any large solution of −u + g(u) = 0 in a ball is radially symmetric. 


Keywords: Elliptic equations; Boundary blow-up; Keller–Osserman condition; Radial symmetry; Spherical Laplacian 


Let BR denote the open ball of center 0 and radius R>0inRN , N 2. A classical result 

due to Gidas, Ni and Nirenberg [9] asserts that, if g is a locally Lipschitz continuous real-valued 

function, any u ∈ C2 (Ω) which is a positive solution of 

 

−u + g(u) = 0 inBR, 

(1.1) 

u = 0 on∂BR 

is radially symmetric. The proof of this result is based on the celebrated Alexandrov–Serrin 

moving plane method [17]. Later on, this method was used in many occasions, with a lot of 


E-mail address: veronl@lmpt.univ-tours.fr (L. Véron). 

1 The author acknowledges the support of RTN European project FRONTS-SINGULARITIES, RTN contract HPRN- 

CT-2002-00274. 


doi:10.1016/j.jfa.2006.03.010

582 A. Porretta, L. Véron / Journal of Functional Analysis 236 (2006) 581–591 

refinements for obtaining selected symmetry results and a priori estimates for solutions of semilinear 

elliptic equations. If the boundary condition is replaced by u = k ∈ R, clearly the radial 

symmetry still holds if u − k does not change sign in BR. Starting from this observation, it was 

conjectured by Brezis [5] that any solution u of 

 

−u + g(u) = 0 inBR, 

(1.2) 

lim|x|→R u(x) =∞ 

is indeed radially symmetric. Notice that this problem admits a solution (usually called a “large 

solution”) if and only if g satisfies the Keller–Osserman condition [10,14] g h on [a,∞), for 

some a>0 where h is non decreasing and satisfies 

∞ 

a 

s 

ds 

√ < ∞, where H(s)= h(t) dt. (1.3) 

H(s) 

Up to now, at least to our knowledge, only partial results were known concerning the radial 

symmetry of solutions of (1.2): in [13], the authors prove this result assuming (besides the Keller– 

Osserman condition) that g ′ (s)/ √ G(s) →∞as s →∞, or for the special case when g(s) = s q , 

using the estimates for the second term of the asymptotic expansion of the solution near the 

boundary. Of course, the symmetry can also be obtained via uniqueness, however uniqueness is 

known under an assumption of global monotonicity and convexity [11,12]. Otherwise, it is easy 

to prove, by a one-dimensional topological argument, that uniqueness for problem (1.2) holds 

for almost all R>0 under the mere monotonicity assumption. However, if g is not monotone, 

uniqueness may not hold (see e.g. [1,13,15]), and it turns out to be very important to know 

whether all the solutions constructed in a ball are radially symmetric, a fact that would lead to 

a full classification of all possible solutions. Let us point out that the interest in such qualitative 

properties of large solutions has being raised in the last few years from different problems (see 

e.g. [1,6–8] and the references therein). 

In this article we prove that Brezis’ conjecture is verified under an assumption of asymptotic 

convexity upon g, namely we prove 

Theorem 1.1. Let g be a locally Lipschitz continuous function. Assume that g is positive and 

convex on [a,∞) for some a>0, and satisfies the Keller–Osserman condition. Then any C 2 

solution of (1.2) is radially symmetric and increasing. 

Notice that the Keller–Osserman condition implies that the function g is superlinear at infinity. 

The convexity assumption on g is then very natural in such context. 

In order to prove Theorem 1.1, we prove first a suitable adaptation of Gidas–Ni–Nirenberg 

moving plane method to the framework of large solutions, without requiring any monotonicity 

assumption on g. This first result, which can have an interest in its own, reads as follows. 

Theorem 1.2. Assume that g is locally Lipschitz continuous and let u be a solution of (1.2) which 

satisfies 

 

lim|x|→R ∂ru =∞, 

(1.4) 

|∇τ u|=o(∂ru) as |x|→R, ∀τ ⊥ x such that |τ|=1, 

a

A. Porretta, L. Véron / Journal of Functional Analysis 236 (2006) 581–591 583 

where ∂ru and ∇τ u are respectively the radial derivative and the tangential gradient of u. Then 

u is radially symmetric and ∂ru>0 on BR \{0}. 

Thus, in view of the previous statement, our main point in order to deduce the general result 

of Theorem 1.1 is to prove that condition (1.4) always holds (even in a stronger form) if we 

assume that g is asymptotically convex, and this is achieved by providing sharp informations on 

the radial and the tangential behaviour of u near the boundary. 

2. Proof of the results 

Let B ={e1,...,eN } be the canonical basis of RN .IfP ∈ RN and ρ>0, we denote by 

Bρ(P ) the open ball with center P and radius ρ, and for simplicity Bρ(0) = Bρ. We consider the 

problem 

 

−u + g(u) = 0 inBR, 

(2.1) 

u(x) =∞ on ∂BR, 

where R>0. By a solution of (2.1), we mean that u ∈ C 2 (BR) is a classical solution in the 

interior of the ball and that u(x) tends to infinity uniformly as |x| tends to R. 

We shall consider the following assumptions on g: 

and satisfies 

g : R → R is locally Lipschitz continuous, (2.2) 

∃a >0 such that g is positive and convex on [a,∞), (2.3) 

 

+∞ 

a 

t 

1 

√ dt < +∞, where G(t) = g(s)ds. (2.4) 

G(t) 

Note that convexity and (2.4) imply that g is increasing on [b,∞) for some b>0. 

If u ∈ C 1 (BR) we denote by ∂u/∂r(x) =〈Du(x), x/|x|〉 the radial derivative of u, and by 

∇τ u(x) = (Du(x) −|x| −1 ∂u/∂r(x))x the tangential gradient of u. Our first technical result, 

which is a reformulation in the framework of large solutions of the famous original proof of 

Gidas, Ni and Nirenberg [9], is the following. 

Theorem 2.1. Assume that g satisfies (2.2), and let u be a solution of (2.1). If there holds 

(ii) 

then u is radially symmetric and ∂u/∂r > 0 in BR \{0}. 

a 

∂u 

(i) lim (x) =∞, 

|x|→R ∂r 

 

∇τ (x) 

 

∂u 

= o 

∂r (x) 

 

as |x|→R, (2.5)


Proof. Since the equation is invariant by rotation, it is sufficient to prove that (2.5) implies that 

u is symmetric in the x1 direction. 

We claim first that for any P ∈ ∂B + := ∂BR ∩{x ∈ R N : x1 > 0}, there exists δ ∈ (0,R)such 

that 

∂u 

(x) > 0 ∀x ∈ BR ∩ Bδ(P ). (2.6) 

∂x1 

Indeed, thanks to (2.5) we have, 

∂u 

= 

∂x1 

∂u x1 

∂r |x| + 

 

Du − ∂u 

 

x 

· e1 

∂r |x| 

= ∂u 

 

x1 

∂r |x| + 

 

∂u 

−1 

Du − 

∂r 

∂u 

 

x 

· e1 

∂r |x| 

= ∂u 

 

x1 

+ o(1) as |x|→R. 

∂r |x| 

Since P ∈ ∂B + , the claim follows straightforwardly. 

Next we follow the construction in [9]. For any λ 0 

∂x1 

inUε ∩ BR. (2.8) 

By definition of μ there holds u uμ in Σμ; thus, if we denote Dε = BR−ɛ/2 ∩ Σμ, wehave 

 

(u − uμ) = a(x)(u − uμ) 

u − uμ 0 

in Dε, 

inDε, 

(2.9)


where a(x) = g(u) − g(uμ)/(u − uμ). Thanks to (2.2) and since Dε is in the interior of BR, 

a(x) is a bounded function in Dε, and the strong maximum principle applies to (2.9). Since u 

tends to infinity at the boundary and is finite in the interior, for ε small we clearly have u ≡ uμ 

in Dε: therefore we conclude that u>uμ in Dε, and, since u = uμ on Tμ ∩ ∂Dε and ∂uμ/∂x1 = 

−∂u/∂x1 on Tμ, it follows from Hopf boundary lemma that 

∂u 

> 0 onTμ∩∂Dε. ∂x1 

Since u ∈ C 1 (BR), the last assertion, together with (2.8), implies that there exists σ>0 such that 

∂u 

> 0 

∂x1 

inBR∩{x: μ − σuμ in Σμ. On the other 

hand, we can also exclude that ¯x ∈ Tμ; indeed, we have 

u(xn) − u 

(xn)λn = 2(xn − λn) ∂u 

(ξn) 

∂x1 

for a point ξn ∈ ((xn)λn ,xn). If (a subsequence of) xn converges to a point in Tμ, then for n large 

we have dist(ξn,Tμ)0 contradicting (2.11). We 

are left with the possibility that ¯x ∈ ∂Σμ \ Tμ: but this is also a contradiction since u blows up at 

the boundary and it is locally bounded in the interior, so that u(xn) − u((xn)λn ) would converge 

to infinity. 

Thus μ = 0 and (2.7) holds in the whole {x ∈ BR: x1 > 0}. We deduce that u is symmetric 

in the x1 direction and ∂u/∂x1 > 0. Applying to any other direction we conclude that u is radial 

and ∂u/∂r > 0. ✷ 

Remark 2.1. Let us recall that in some special examples (for instance when g(s) has an exponential 

or a power-like growth) the asymptotic behaviour at the boundary of the gradient of the 

large solutions has already been studied (see e.g. [2,4,16]) so that the previous result could be 

directly applied to prove symmetry. In general, through a blow-up argument, we are able to prove 

(2.5) if 

s ↦→ g(s) 

√ G(s) 

∞ 

s 

1 

√ 2G(ξ) dξ (2.12)


is bounded at infinity; however, this assumption does not include the case when g has a slow 

growth at infinity (such as g(r) ≡ r(ln r) α with α>2) and is not so general as (2.3). 

Theorem 2.2. Assume that (2.2)–(2.4) hold. Then any solution u of (2.1) is radial and ∂u/∂r > 0 

in BR \{0}. 

The following preliminary result is a consequence of more general results in [3,11,12,18]. 

However, we provide here a simple self-contained proof for the radial case. 

Lemma 2.1. Let h be a convex increasing function satisfying the Keller–Osserman condition 

 

+∞ 

a 

s 

ds 

√ < ∞, H(s)= h(t) dt, (2.13) 

H(s) 

for some a>0. Then the problem 

 

−v + h(v) = 0 in BR, 

lim|x|→R v(x) =∞ 

has a unique solution. 

a 

(2.14) 

Proof. Since h is increasing, there exist a maximal and a minimal solution v and v, which are 

both radial, so that it is enough to prove that v = v. To this purpose, observe that if v is radial 

we have (v ′ rN−1 ) ′ = rN−1 h(v) so that, since v ′ (0) = 0, and replacing H by H ˜ = H − H(min v) 

which is nonnegative on the range of values of v,wehave 

which yields 

Define now 

(v ′ r N−1 ) 2 

2 

 

= 

0 

r 

s 2(N−1) h(v)v ′ ds r 2(N−1) ˜ 

H(v) 

0 v ′ 

< 2H(v). ˜ 

(2.15) 

 

w = F(v)= 

v 

∞ 

ds 

. 

2H(s) ˜ 

A straightforward computation, and condition (2.13), show that w solves the problem 

 

w = b(w)(|Dw| 2 − 1) in BR, 

w = 0 on∂BR, 

(2.16)


 

where b(w) = h(v)/ 2H(v). ˜ One can easily check that the convexity of h implies that 

 

h(v)/ 2H(v)is ˜ nondecreasing, hence b(w) is nonincreasing with respect to w. Moreover, since 

|Dw|=|w ′ |=v ′ 

/ 2H(v), ˜ from (2.15) one gets |w ′ | < 1. Note that the transformation v ↦→ w 

establishes a one-to-one monotone correspondence between the large solutions of (2.14) and 

the solutions of (2.16), so that w = F(v) and w = F(v) are respectively the minimal and the 

maximal solutions of (2.16). Thus we have 

′ N−1 

(w − w ) r ′ N−1 

= r b(w) |w ′ | 2 − 1 − b(w ) |w ′ | 2 − 1 r N−1 b(w) |w ′ | 2 −|w ′ | 2 , 

so that the function z = (w − w ) ′ r N−1 satisfies 

z ′ a(r)z, z(0) = 0, where a(r) = b(w)(w ′ + w ′ ). 

Because a is locally bounded on [0,R), we deduce that z 0, hence w − w is nondecreasing. 

Since w − w is nonnegative and w(R) = w(R) = 0 we deduce that w = w, hence v = v. ✷ 

Lemma 2.2. Assume that g satisfies (2.3) and (2.4), and that u is a solution of (2.1). Then 

and the two limits hold uniformly with respect to {x: |x|=r}. 

(i) lim 

|x|→R ∇τ u(x) = 0, 

(ii) 

∂u 

lim (x) =∞, 

|x|→R ∂r 

(2.17) 

Proof. In spherical coordinates (r, σ ) ∈ (0, ∞) × S N−1 the Laplace operator takes the form 

u = ∂2u N − 1 ∂u 

+ 

∂r2 r ∂r 

1 

+ su, 

r2 where s is the Laplace–Beltrami operator on SN−1 .If{γj } N−1 

j=1 

on SN−1 crossing orthogonally at ˜σ = γj (0), there holds 

su(r, ˜σ)= 

j1 

d 2 u(r, γj (t)) 

dt 2 

 

 

 

t=0 

is a system of N − 1 geodesics 

. (2.18) 

On the sphere the geodesics are large circles. The system of geodesics can be obtained by considering 

a set of skew symmetric matrices {Aj } N−1 

j=1 such that 〈Aj ˜σ,Ak ˜σ 〉=δk j , and by putting 

γj (t) = etAj ˜σ . 

Step 1. Two-side estimate on the tangential first derivatives. 

By assumption (2.3) g can be written as 

g(s) = g∞(s) +˜g(s),


where g∞(s) is a convex increasing function satisfying (2.4) and ˜g(s) is a locally Lipschitz 

function such that ˜g ≡ 0in[M,∞) for some M>0. In particular, u satisfies 

u − g∞(u) =˜g(u). 

Since u blows up uniformly, there holds u(x) M if |x|∈[r0,R)for a certain r0 2, 

ln r−ln R 

ln r0−ln R if N = 2, 

and v h (x) = u h (x) +|h|LP (|x|); then v h = u h , and since g∞ is increasing, 

(2.22) 

v h − u h 

g∞ v − g∞(u) in BR \ Br0 . (2.23)


Observe that u h ,asu, also satisfies (2.21), so that in particular 

u h (x) − u(x) → 0 as|x|→R. (2.24) 

Therefore vh (x)−u(x) → 0as|x|→R too, while by construction vh u on ∂Br0 . We conclude 

from (2.23) (e.g. using the test function (vh − u + ε)−, which is compactly supported, and then 

letting ε go to zero) that 

v h = u h +|h|LP (r) u. 

We recall that the Lie derivative LAj u of u(r, ·) following the vector field tangent to SN−1 η ↦→ 

Aj η is defined by 

so we get, by letting h → 0, 

LAj u(r, σ ) = du(r,etAj σ) 

dt 

 

 

 

t=0 

 

LAj u(r, ˜σ) LP (r) < C(R − r). (2.25) 

Step 2. One-side estimate on the tangential second derivatives. 

Next we define the function w h by 

w h = uh + u−h − 2u 

h2 . 

As before, let r0 0 such that 

w h ˜L on ∂Br0 . 

Moreover, from (2.25) we get that wh = 0on∂BR. We conclude that 

h 

w 

+ (x) ˜LP |x| , 

,


where P(r)is defined in (2.22). Letting h tend to zero we obtain 

d 2 u(r, e tAj ˜σ) 

dt 2 

 

 

 

t=0 

Using (2.18), and the fact that ˜σ is arbitrary, we derive 

˜LP (r) for r ∈[r0,R). (2.26) 

su(r, σ ) (N − 1) ˜LP(r) ∀(r, σ ) ∈[r0,R)× S N−1 . (2.27) 

Step 3. Estimate on the radial derivative. 

Using (2.22) and (2.27) we deduce that 

Therefore 

 

∂ 

r 

∂r 

(su) + (x) = o(1) uniformly as |x|→R. 

 

N−1 ∂u 

= r 

∂r 

N−1 

 

g(u) − 1 

su 

r2 

r N−1 g∞(u) − o(1) 

uniformly as r → R. (2.28) 

Now one can easily conclude: let z(r) denote the minimal (hence radial) solution of 

⎧ 

⎨ z = g∞(z) in BR \ Br0 

⎩ 

, 

z = min∂Br u 

0 

on ∂Br0 , 

limr→R z =∞. 

We have u(x) z(x) if |x| ∈[r0,R), hence g∞(u) g∞(z). Because this last function is not 

integrable near ∂BR, one obtains 

lim 

r 

r→R 

r0 

Clearly (2.28) implies 

s N−1 

g∞ u(s, σ ) ds →∞ uniformly for σ ∈ SN−1 . 

∂u r→R 

(r, σ ) −−−→∞ 

∂r 

uniformly for σ ∈ SN−1 . 

This completes the proof of (2.17). ✷ 

Proof of Theorem 2.2. By assumptions (2.3) and (2.4), and Lemma 2.2, we deduce that u satisfies 

(2.5), hence we apply Lemma 2.1 to conclude. ✷ 

Finally, let us point out that thanks to Lemma 2.2 and using the moving plane method as in 

Theorem 2.1, we can derive a result describing the boundary behaviour of any solution of 

 

−u + g(u) = 0 inΓR,r ={x ∈ RN : r


which extends a similar result in [9]. 

Corollary 1. Assume that g satisfies (2.2)–(2.4). Then any solution of (2.29) satisfies (2.17) and 

verifies ∂ru>0 on ΓR,(R+r)/2. 

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Non-structural controllability of linear elastic systems 

with structural damping 

Luc Miller a,b,∗ 

a Équipe Modal’X, EA 3454, Université Paris X, Bât. G, 200 Av. de la République, 92001 Nanterre, France 

b Centre de Mathématiques Laurent Schwartz, UMR CNRS 7640, École Polytechnique, 91128 Palaiseau, France 




Abstract 

This paper proves that any initial condition in the energy space for the plate equation with square root 

damping ¨ζ − ρ˙ζ + 2ζ = u on a smooth bounded domain, with hinged boundary conditions ζ = ζ = 0, 

can be steered to zero by a square integrable input function u supported in arbitrarily small time interval 

[0,T] and subdomain. As T tends to zero, for initial states with unit energy norm, the norm of this u 

grows at most like exp(Cp/T p ) for any real p>1andsomeCp > 0. Indeed, this fast controllability 

cost estimate is proved for more general linear elastic systems with structural damping and non-structural 

controls satisfying a spectral observability condition. Moreover, under some geometric optics condition on 

the subdomain allowing to apply the control transmutation method, this estimate is improved into p = 1 

and the dependence of Cp on the subdomain is made explicit. These results are analogous to the optimal 

ones known for the heat flow. 


Keywords: Controllability; Observability; Control cost; Plates; Structural damping; Transmutation 

A wide variety of dissipative linear elastic control systems may be represented by a secondorder 

differential equation in a Hilbert space: 

* Fax:+330169333019. 

E-mail address: miller@math.polytechnique.fr. 

¨ζ(t)+ D ˙ζ(t)+ Sζ(t) = Bu(t), t ∈ R+, (1) 


doi:10.1016/j.jfa.2006.03.001

L. Miller / Journal of Functional Analysis 236 (2006) 592–608 593 

where each dot denotes a derivative with respect to the time variable t and the function ζ represents 

the evolution of the system under the action of the input function u.Thestructural vibration 

modes of the conservative system represented by (1) with B = D = 0 are prescribed by the positive 

self-adjoint operator S. This ideal system is perturbed by a dissipative mechanism prescribed 

by the positive self-adjoint operator D. The system is actuated through a control mechanism prescribed 

by the operator B (possibly unbounded to take into account trace operators prescribing 

the boundary value of distributed states). Throughout this paper, controllability will always mean 

the ability of steering any initial state (z(0), ˙z(0)) to zero over a finite time by some appropriate 

input function u (i.e. exact controllability to zero or null controllability). 

This paper concerns the specific dissipative mechanism D = S α with α ∈ (0, 1) called structural 

damping, which generalizes the square root damping model α = 1/2 introduced in [6]: 

“The basic property of structural damping, which is said to be consistent with empirical studies, 

is that the amplitudes of the normal modes of vibration are attenuated at rates which are proportional 

to the oscillation frequencies.” This model was also studied under the name “proportional 

damping” (cf. [3]). The quite different case α = 1 is known as “Kelvin–Voigt” damping. When 

B is the identity and α ∈ (0, 1], this is the first class of parabolic-like control models considered 

in [13,24] with the extra assumption that S has compact resolvent, dispensed with in [2]. 

This paper focuses on the cost of fast controls as in [2,24]. The controllability results known 

for these systems hold for a control time which can be chosen as small as wished. This asymptotic 

is referred to as fast control. Thecost over a given time is the supremum over every initial state 

with unit energy norm of the smallest norm of an input function which steers it to zero over the 

given time. The study of the cost of fast controls was initiated by Seidman (cf. references in [17]) 

and recently revived by Da Prato who connected it to some properties of stochastic differential 

equations (cf. references in [2]). 

The earlier results restricted to elementary forms of control operators (mainly B is the identity 

or has rank one, cf. [2,11,13,14,21,23,24]). A key point in their proofs is (loosely speaking) the 

existence of a common eigenbasis for the three operators S, D and B, modeling respectively the 

structure, the damping and the control. On the contrary, the controllability results of this paper 

apply to non-structural controls, e.g. locally distributed control. 

The main application is to the plate equation with square root damping on a smooth bounded 

domain M of R d with hinged boundary conditions: 

¨ζ − ρ˙ζ + 2 ζ = u on R+ × M, ζ = ζ = 0 onR+ × ∂M, (2) 

where ρ>0 and the input function u is supported on a non-empty subdomain Ω (cf. Theorem 2). 

Fast controllability is proved to hold for any control region Ω. As the control time T tends to 

zero, the cost is proved to grow at most like exp(Cβ/T β ) for any β>1 and some Cβ > 0. If 

the length LΩ of the longest generalized ray of geometrical optics in M which does not intersect 

Ω is finite (this is the condition of [5]) and ρLΩ and some positive C which does not depend on Ω. These results 

are analogous to the optimal fast controllability cost known for the heat flow (cf. [10,17]). They 

confirm the formal analogy: ∂ 2 t −2∂t + 2 = (∂t −) 2 . On the contrary, when Ω = M (i.e. B is 

the identity), [2,24] prove that the fast controllability cost grows like 1/T β for some β 1/2, as 

in finite-dimensional systems (cf. [22]). 

Earlier methods to estimate the cost of fast controls were global parabolic Carleman estimates 

(cf. [2,10]), the Fourier transform method for constructing functions bi-orthogonal to exponential 

series (cf. [17,23] and references therein) and the transmutation control method (cf. [17,19]). The

594 L. Miller / Journal of Functional Analysis 236 (2006) 592–608 

last two are combined in Section 2 to take the geometry of the control region into account and 

improve the cost estimate for (2) as stated above and more precisely in Theorem 2. 

The proof of the abstract result, Theorem 1 of Section 1, applies a new method using the control 

strategy of Lebeau and Robbiano in [15] as implemented in [16] (the companion paper [20] 

applies this method to a simpler model: the holomorphic semigroup generated by exp(−tS β ), 

β>0). The key assumption is an observability condition on the spectral subspaces of S with 

respect to B stated in Definition 1. It is an abstract version of a result on sums of eigenfunctions 

of the Dirichlet Laplacian proved in [12,16] by local elliptic Carleman estimates and extended to 

non-compact manifolds in [18]. Therefore the abstract result applies to (2) (such concrete models 

with other forms of controls are considered e.g. in [14, Chapter 3]) even if, e.g., M is unbounded, 

the Dirichlet Laplacian is positive with non-compact resolvent and Ω is the exterior of a compact 

subdomain. 

It is clearly desirable to study plate equations with other boundary conditions or with locally 

distributed controls on the boundary instead of the interior. Other open problems are mentioned 

in the remarks of Section 2.1. 

1. A structurally damped linear elastic control system 

Before stating the abstract model and the theorem precisely, we need to introduce a few notations. 

Let H0 and U be Hilbert spaces with respective norms ·0 and ·.LetA be a self-adjoint, 

positive and boundedly invertible unbounded operator on H0 with domain D(A). We introduce 

the Sobolev scale of spaces based on A. For any positive integer p, letHp denote the Hilbert 

space D(A p/2 ) with the norm xp =A p/2 x0 (which is equivalent to the graph norm x0 + 

A p/2 x0). We identify H0 and U with their duals. Let H−p denote the dual of Hp. Since Hp is 

densely continuously embedded in H0, the pivot space H0 is densely continuously embedded in 

H−p, and H−p is the completion of H0 with respect to the norm x−p =A −p/2 x0. 

Let the observation operator C be in L(H2,U), which denotes bounded operators from H2 

to U, and let B ∈ L(U, H−2) denote the dual of C. 

Let α ∈ (0, 1) denote the structural dissipation power, and let ρ>0 denote the dissipativity 

constant. With the structural operator A and the control operator B, they define the second-order 

Cauchy problem with input function u: 

¨ζ(t)+ ρA 2α ˙ζ(t)+ A 2 ζ(t)= Bu(t), 

ζ(0) = ζ0 ∈ H2, ˙ζ(0) = ζ1 ∈ H0, u∈L 2 loc (R; U). (3) 

In order to define the (mild) solution of this problem, we assume that B ∈ L(U, H0) (which 

is enough for the application in Section 2) or, more generally, that B is admissible in a sense 

specified later in (9). 

To state the “observability condition” on the spectral subspaces of A with respect to C of 

the main theorem, we first introduce our spectral notations. Given γ>0 and μ>1, applying 

the functional calculus for self-adjoint operators to the positive operator A γ and the bounded 

function on R + defined by 1λμ = 1ifλ μ and 1λμ = 0, otherwise, yields the spectral 

projector 1A γ μ. The image of H0 under this projection operator is just the spectral subspace 

1A γ μH0 of A γ .


Definition 1. Let γ>0. The observability of low modes of A γ through C at exponential cost 

holds if there are positive constants D0 and D1 such that 

∀μ>1, ∀v ∈ 1A γ μH0, v0 D0e D1μ Cv. (4) 

This abstract condition is satisfies in some concrete applications given in the next section. As 

illustrated in the proof of the following main theorem (cf. Section 1.3), it allows to compare the 

free dissipation of high modes to the cost of controlling low modes. 

Theorem 1. Assume that observability of low modes of A γ through C at exponential cost 

holds for some γ ∈ (0, 1) (cf. Definition 1). For all ρ>0 and α ∈ (γ /2, 1 − γ/2), for all 

β>(2min{α, 1 − α}/γ − 1) −1 , there are positive constants C1 and C2 such that for all 

T ∈ (0, 1], for all ζ0 and ζ1, there is an input function u such that the solution ζ of (3) satisfies 

ζ(T)= ˙ζ(T)= 0 with the cost estimate: 

T 

0 

 

u(t) 2 dt C2 exp 

 

C1 

T β 

1.1. The duality between observation and control 

 

ζ0 2 2 +ζ1 2 0 . 

The proof of Theorem 1 uses the well-known equivalence between controllability and observability 

(cf. [9]). In this section, we clarify in what sense the dual of the control problem (3) is the 

observation of the following Cauchy problem (without input): 

¨z(t) + ρA 2α ˙z(t) + A 2 z(t) = 0, z(0) = z0 ∈ H2, ˙z(0) = z1 ∈ H0. (5) 

The second-order differential equations (3) and (5) may be restated as first-order systems by 

setting ξ(t) = (ζ(t), ˙ζ(t)) and x(t) = (z(t), ˙z(t)): 

˙ξ(t)− Aξ(t) = Bu(t), ξ(0) = ξ0 ∈ X, u ∈ L 2 loc (R; U), (6) 

˙x(t) = Ax(t), x(0) = x0 ∈ X. (7) 

The state space is X = H2 × H0. The semigroup generator A of (7) is defined by 

 

0 I 

A = 

−A2 −ρA2α 

, D(A) = (z0,z1) ∈ H2 × H2 | A 2 z0 + ρA 2α 

z1 ∈ H0 . 

It inherits from −A the necessary and sufficient properties of Lumer–Phillips for generating a 

contraction semigroup. 

The control operator is B = ΠB, where Π : X → H0 is defined by Π(z0,z1) = z1. IfB ∈ 

L(U, H0) (as in the application in Section 2) then B ∈ L(U, X). Indeed Theorem 1 is valid in 

the following more general (canonical) setting introduced by Weiss in [25]. Let X1 be D(A) 

with the norm x1 =Ax and let X−1 be the completion of X with respect to the norm 

x−1 =A −1 x. At first, we only assume B ∈ L(U, X−1). In order to define the unique (mild) 

solution ξ ∈ C(R+; X) of (6) by the integral formula


ξ(t) = e tA t 

ξ(0) + e (t−s)A Bu(s) ds, (8) 

0 

we also make the admissibility assumption: forsomeT>0 (hence for all T>0) there is a 

positive constant KT such that 

∀u ∈ L 2 loc (R; U), 

 

T 

 

e 

 

tA 

 

 

Bu(t) dt 

 

0 

2 

KT 

We define the duality pairing on X by 

 

(ζ0,ζ1), (z0,z1) =〈Aζ0,Az0〉0 −〈ζ1,z1〉0. 

T 

0 

 

u(t) 2 dt. (9) 

With respect to this pairing, X and A are their own dual, X−1 is the dual of X1 and B is the dual 

of the observation operator C = CΠ. The assumptions on B are equivalent to C ∈ L(X1,U)and, 

for all x0 ∈ D(A), 

T 

0 

 

tA 

Ce x0 

2 dt KT x0 2 . 

Therefore the output map x0 ↦→ Ce tA x0 from D(A) to L 2 ([0,T]; U) has a continuous extension 

to X.N.b.ifα 1/2, then X1 = H4 × H2 and X−1 = H0 × H−2. 

We recall the duality between controllability and observability (cf. [9]). 

Lemma 1. Let T>0 and CT > 0. The following properties are equivalent: 

(i) For all initial state ξ0 ∈ X, there is an input function u ∈ L2 loc (R; U) such that the solution 

ξ ∈ C(R+; X) of (6) satisfies ξ(T) = 0 and uL2 (0,T ;U) CT ξ0. 

(ii) For all initial state x0 ∈ X, the solution x(t) = etAx0 of (7) satisfies the observation inequality 

x(T ) CT Cx(t)L2 (0,T ;U) . 

N.b. the smallest constant CT such that these properties hold is the controllability cost mentioned 

in the introduction. The estimate in Theorem 1 writes C 2 T C2 exp(C1/T β ). The contractivity 

of e tA , (8) and (9) imply the estimates: 

T 

0 

 

ξ(T) 2 2 1 + KT C 2 

ξ(0) T 

2 , (10) 

 

ξ(t) 2 dt 2T 1 + KT C 2 

ξ(0) T 

2 , (11) 

since Kt and t 

0 u(s)2 ds are nondecreasing, although Ct is nonincreasing.

1.2. Spectral and growth bounds 


The proof of Theorem 1 relies on a spectral decomposition of the problem. We extend the 

action of the spectral projector 1A γ μ to X according to 1A γ μ(z0,z1) = (1A γ μz0, 1A γ μz1). 

It commutes with A and the generated semigroup. Therefore X is the orthogonal sum of the 

invariant subspaces 1A γ μX (low modes) and 1A γ >μX (high modes). 

The restriction of e tA to 1A γ >μX satisfies the following exponential decay bound. 

Proposition 1. Let γ ′ = γ/(2min{α, 1 − α}). There is an r>0 such that 

∀μ 1, ∀x ∈ 1A γ >μX, ∀t 0, 

 

e tA x exp 1/γ ′ 

−rμ t x. 

This reduces to a spectral bound thanks to the results of Chen and Triggiani on the differentiability 

of e tA (cf. [7,8]). We first prove two spectral lemmas. 

Lemma 2. The spectrum of A relates to the spectrum of A according to 

σ(−A) ⊂ λ ∈ C |∃μ ∈ σ(A), Pμ(λ) = 0 with Pμ(λ) = λ 2 − ρμ 2α λ + μ 2 . 

Proof. Let λ/∈{λ ∈ C |∃μ ∈ σ(A), Pμ(λ) = 0}. The function μ ↦→ Pμ(λ)/μ2 is continuous 

on (0, +∞) ⊃ σ(A), it tends to 1 as μ tends to infinity and it does not vanish on the 

closed set σ(A). Hence, there is an ε>0 such that, for all μ ∈ σ(A), |Pμ(λ)/μ2 | >ε. Since 

μ ↦→|μ2Pμ(λ) −1 | is bounded on σ(A) (by ε−1 ), we have A2PA(λ) −1 ∈ L(H0) ⊂ L(H2,H0), 

PA(λ) −1 ∈ L(H0,H4) ⊂ L(H0,H2) and (λI − ρA2α )PA(λ) −1 ∈ L(H2). Therefore the operator 

M(λ) defined by 

 

(λI − ρA2α )PA(λ) 

M(λ) = 

−1 −PA(λ) −1 

A 2 PA(λ) −1 λPA(λ) −1 

is bounded on X.ButM(λ)(λI + A) = (λI + A)M(λ) = I , so that M(λ) is the bounded inverse 

of λI + A. Hence λ/∈ σ(−A). ✷ 

Lemma 3. The roots λ± = (1 ± 1 − (2μ 1−2α /ρ) 2 )ρμ 2α /2 of Pμ(λ) satisfy: 

∀μ 1, min{Re λ+, Re λ−} rμ 2min{α,1−α} , with r = min{ρ/2, 1/ρ}. 

Proof. Let x = 2μ 1−2α /ρ. Ifx 1, then Re λ+ = Re λ− = μ 2α ρ/2. Otherwise λ± ∈ R, λ+ = 

(1 + √ ...)μ 2α ρ/2 μ 2α ρ/2, and λ− = (1 − √ 1 − x 2 )μ 2α ρ/2 x 2 μ 2α ρ/4 = μ 2(1−α) /ρ. 

Since min{μ 2α ,μ 2(1−α )}=μ 2min{α,1−α} for μ 1, gathering these lower bounds yields the 

lemma. ✷ 

Proof of Proposition 1. Since etA is a differentiable semigroup for α ∈ (0, 1] ([8] proves that 

this semigroup is of Gevrey class and that it is analytic if and only if α ∈[1/2, 1]), it is eventually 

continuous for the operator norm topology. The semigroup generated by the restriction Aμ of A 

to 1Aγ >μX inherits this property. But Lemma 3 and the proof of Lemma 2 imply σ(−Aμ) ⊂ 

{λ ∈ C | Re λ rμ1/γ ′ 

} with r = min{ρ/2, 1/ρ}. Therefore the growth bound in Proposition 1 

holds. ✷


1.3. Proof of Theorem 1 

The last ingredient of this proof is the following cost estimate corresponding to the control 

operator B = I proved in [2]. 

Proposition 2. (Avalos–Lasiecka, 2003) For all ρ>0, for all α ∈ (0, 1), there are positive constants 

c1 and c2 such that for all T ∈ (0, 1] the solutions of (5) satisfy: 

∀z0 ∈ H2, ∀z1 ∈ H0, 

 

z(T ) 2 2 + ˙z(T ) 2 c2 

0 T c1 

T 

0 

 

˙z(t) 2 dt. (12) 

(Indeed, [2] specifies how the power c1 depends on α.) 

In a first step, from the stationary condition in Definition 1, Proposition 2 and the duality in 

Lemma 1, we deduce the “controllability of low modes at exponential cost” in the corresponding 

dynamics. In a second step, combining it with the decay bound in Proposition 1 according to the 

iterative control strategy introduced by Lebeau and Robbiano in [15], we prove the controllability 

of all modes. We estimate the controllability cost as the control time tends to zero, like in [20], 

in the last step. 

First step. With the notations introduced in Section 1.1, the observation inequality (12) in Proposition 

2 writes: 

∀x0 ∈ X, 

 

e T A x0 

 

2 c2 

T 

T c1 

0 

 

tA 

Πe x0 

2 dt. (13) 

Let τ ∈ (0, 1], μ 1 and x0 ∈ 1A γ μX. For all t ∈[0,τ], we may apply (4) to Πe tA x0 since it 

is in 1A γ μH0: 

 

Πe tA 

x02 

0 D2 0e2D1μ CΠe tA 

x02 

. 

First integrating on [0,τ], then using (13) yields: 

 

e T A x0 

 

2 D 2 0 

c2 

e2D1μ 

τ c1 

τ 

0 

 

Ce tA 

x02 

dt. 

This “low modes fast observability for e tA at exponential cost” is equivalent, by the same duality 

as in Lemma 1, to the controllability property: for all τ ∈ (0, 1] and μ>1, there is a bounded 

operator S τ μ : X → L2 (0,τ; U) such that, for all ξ0 ∈ 1A γ μX, the solution ξ ∈ C(R+,X) of 

(6) with input function u = S τ μ ξ0 satisfies 1A γ μξ(τ) = 0, and ∃d3 > 0, S τ μ (d3/τ c1/2 )e D1μ 

(cost estimate). 

Second step. The hypothesis on α implies that the γ ′ of Proposition 1 is lower than 1. We 

introduce a dyadic scale of modes μk = 2 k (k ∈ N) and a sequence of time intervals τk = σδT/μ δ k


where δ ∈ (0,γ ′−1 − 1) and σδ = (2 

k∈N 2−kδ ) −1 > 0, so that the sequence of times defined 

recursively by T0 = 0 and Tk+1 = Tk + 2τk converges to T . The strategy consists in steering the 

initial state ξ0 to 0, through the sequence of states ξk = ξ(Tk) ∈ 1Aγ >μk−1X composed of ever 

higher modes, by applying recursively the input function uk = S τk 

μkξk to ξk during a time τk and 

no input during a time τk. Introducing the notations 

εk =ξk, Ck = D2e D1μk /τ c1/2 

k , and ρk = 

the cost estimate of the previous step writes S τk 

μk Ck and implies: 

u 2 

L 2 (0,T ;U) 

= 

k∈N 

uk 2 

L 2 (0,τk;U) 

 

k∈N 

Ck+1εk+1 

Ckεk 

Since τk T 1, the estimate (10) between the times Tk and Tk + τk implies 

 

ξ(Tk + τk) 2 2 1 + K1C 2 2 

k εk . 

2 

, (14) 

C 2 k ε2 k . (15) 

Since 1Aγ μkξ(Tk ′ 

−rμ1/γ + τk) = 0 and Proposition 1 imply εk+1 e k 

τkξ(Tk + τk), we deduce 

ε 2 k+1 

1/γ ′ 

2e−2rτkμk 1 + K1C 2 2 

k εk . 

Since Ck+1/Ck = 2 δc1/2 e D1μk , we deduce that, for any D3 > 4D1, there is a D4 > 0 such that 

ρk 2 1+δc1 

Since γ ′−1 − δ>1, this implies: 

 

e −2D1μk 

K1D 

+ 2 2 

τ c1 

 

1/γ ′ 

4D1μk−2rτkμ 

e k 

k 

D4 

T c1 eD3μk−2rσδTμ γ ′−1−δ k . (16) 

∀ρ ∈ (0, 1), ∃N ∈ N, k N ⇒ ρk ρ. 

Therefore limk εk = 0 and the last series in (15) converges. This completes the proof of the 

controllability in Theorem 1. 

Third step. The controllability cost CT , formally defined after Lemma 1, satisfies: 

Since 

l μl, 

 

0kl−1 

C 2 T C2 0 

 

1 + 

 

l1 0kl−1 

μk μl and 

0kl−1 

ρk 

 

. (17) 

μ γ ′−1 −δ 

k 

μ γ ′−1−δ l−1 /2,


(16) implies 

 

0kl−1 

ρk exp D3 + ln D4/T c1 μl − rσδTμ γ ′−1−δ l−1 . 

Hence, setting q = 2 γ ′−1 −δ and T ′ = rσδT/q we have 

∀l 1, 

 

0kl−1 

As in [20], plugging this in (17) yields the cost estimate: 

ρk exp DT ′2l − T ′ q l with DT ′ ∼ 

T ′ →0 c1 ln(1/T ′ ). 

∀β >βq, ∃D6 > 0, ∃D7 > 0, C 2 T D6 

 

D7 

exp 

T ′β 

 

−1 ln q 

with βq = − 1 . 

ln 2 

Since T ′ is proportional to T and βq decreases to (γ ′−1 − 1) −1 = (2min{α, 1 − α}/γ − 1) −1 as 

δ decreases to 0, this proves the estimate in Theorem 1 restated after Lemma 1. 

2. Interior controllability of structurally damped plates 

This section concerns concrete applications of the abstract model studied in the previous section. 

The main application is to the plate equation with square root damping and interior control 

in Ω with hinged boundary conditions: 

¨ζ − ρ˙ζ + 2 ζ = χΩu on R+ × M, ζ = ζ = 0 onR+ × ∂M, 

ζ(0) = ζ0 ∈ H 2 (M) ∩ H 1 0 (M), ˙ζ(0) = ζ1 ∈ L 2 (M), u ∈ L 2 loc (R+ × M). (18) 

In this section, M is a smooth connected complete d-dimensional Riemannian manifold with 

metric g and non-empty boundary ∂M, M denotes the interior and M = M ∪ ∂M.Let denote 

the Dirichlet Laplacian on L 2 (M) with domain D() = H 1 0 (M) ∩ H 2 (M) (thus denotes a 

negative differential operator with variable coefficients depending on the metric g). N.b. the 

results are already interesting when (M, g) is a smooth domain of the Euclidean space R d ,so 

that = ∂ 2 /∂x 2 1 +···+∂2 /∂x 2 d .LetχΩ denote the multiplication by the characteristic function 

of an open subset Ω =∅of M. 

For simplicity, in the following theorem proved in Section 2.4, we assume that M is compact. 

The second part of this theorem makes a geometric assumption on Ω due to Bardos–Lebeau– 

Rauch based on generalized geodesics. In this context, the generalized geodesics are continuous 

trajectories t ↦→ x(t) in M which follow geodesic curves at unit speed in M (so that on these 

intervals t ↦→ ˙x(t) is continuous); if they hit ∂M transversely at time t0, then they reflect as light 

rays or billiard balls (and t ↦→ ˙x(t) is discontinuous at t0); if they hit ∂M tangentially then either 

there exists a geodesic in M which continues t ↦→ (x(t), ˙x(t)) continuously and they branch onto 

it, or there is no such geodesic curve in M and then they glide at unit speed along the geodesic 

of ∂M which continues t ↦→ (x(t), ˙x(t)) continuously until they may branch onto a geodesic 

in M. For this result and whenever generalized geodesics are mentioned, we make the additional 

assumption that they can be uniquely continued at the boundary ∂M (as in [5], to ensure this, we


may assume either that ∂M has no contacts of infinite order with its tangents, or that g and ∂M 

are real analytic). 

Let LΩ denote the length of the longest generalized geodesic in M which does not intersect 

Ω. For instance, we recall that LΩ < ∞ if Ω is a neigbourhood of the boundary of a 

smooth domain M of R d (in that case LΩ is the length of the longest segment in M \ Ω) and 

that LΩ < 2D if Ω is a neighborhood of a hemisphere of the boundary of a Euclidean ball M of 

diameter D. 

Theorem 2. For all ρ>0 and Ω =∅, for all β>1, there are C1 > 0 and C2 > 0 such that, 

for all T ∈ (0, 1], for all ζ0 and ζ1, there is an input function u such that the solution ζ of (18) 

satisfies ζ(T)= ˙ζ(T)= 0 and the cost estimate: 

T 

0 

 

M 

|u| 2 dxdt C2 exp 

C1/T 

β 

M 

|ζ0| 2 +|ζ1| 2 dx. 

For all ρ ∈ (0, 2) and L>LΩ, this result holds with this estimate improved by replacing 

exp(C1/T β ) with exp(CρL 2 /T) where Cρ does not depend on Ω. 

2.1. Application of Theorem 1 

Indeed, Theorem 1 applies to structurally damped plates with interior control in Ω more 

general than (18): 

¨ζ + ρ(−) 2α ˙ζ + 2 ζ = χΩu on R+ × M, 

ζ(0) = ζ0 ∈ H 2 (M) ∩ H 1 0 (M), ˙ζ(0) = ζ1 ∈ L 2 (M), u ∈ L 2 [0,T]×M . (19) 

This is the abstract system (6) where the generator is A =−, the state and input space is 

H0 = U = L 2 (M), and the control and observation operator is B = C = χΩ. N.b. for square root 

damping (α = 1/2), X1 = H4 × H2 ={(z0,z1) ∈ H 4 (M) × H 2 (M) | z1 = z0 = z0 = 0on∂M} 

and the solution of (6) satisfy the boundary conditions ζ = ζ = 0onR+ × ∂M in a generalized 

sense. 

If M is not compact, assume that Ω is the exterior of a compact set K such that K ∩ Ω ∩ 

∂M =∅and that 0 /∈ σ(). In this setting, the observability of low modes of (−) 1/2 through 

C at exponential cost holds (cf. Definition 1). When M is compact this is an inequality on sums 

of eigenfunctions proved as Theorem 3 in [16] and Theorem 14.6 in [12]. This was generalized 

to non-compact M in [18]. Applying Theorem 1 with γ = 1/2 yields: 

Corollary 1. For all ρ>0, α ∈ (1/4, 3/4), and β>min{4α − 1, 3 − 4α} −1 , there are C1 > 0 

and C2 > 0 such that, for all T ∈ (0, 1], for all ζ0 and ζ1, there is an input function u such that 

the solution ζ of (19) satisfies ζ(T)= ˙ζ(T)= 0 and the cost estimate: 

u 2 

L2 C2 exp C1/T β ζ0 2 

H 2 +ζ1 2 

L2 

. 

Remark 1. Note that Proposition 2 proves controllability from Ω = M when α ∈[0, 1). It results 

from [2] that controllability does not hold in Corollary 1 for α = 1. For α = 0, it can be proved


by the transmutation control method that the controllability for ρ = 0 (which holds if Ω satisfies 

the condition LΩ < ∞ of Theorem 2) implies the controllability over the same time for ρ>0. 

The case α ∈ (0, 1/4]∪[3/4, 1) with Ω = M is still open. 

Remark 2. More generally, Theorem 1 applies to A = (−) 1/(2γ) with γ ∈ (0, 1). It does not 

apply to the wave equation which would correspond to γ = 1. The wave equation with square 

root damping (α = 1/2) is just out of reach and seems to us an interesting open problem (the 

appendix in [20] proves that it is not controllable by a one-dimensional input). It results from [1] 

that the wave equation (γ = 1) with Kelvin–Voigt damping (α = 1) is not controllable from 

any Ω = M. It results from [5] that the damped wave equation (γ = 1, α = 0), is controllable 

from Ω or not depending on whether the control time is greater or lower than LΩ (defined before 

Theorem 2). 

2.2. Smoothing 

The control transmutation method of Section 2.4 applies to initial data smoother than in Theorem 

2. This drawback is easily overcome by a general abstract remark, made here, concerning 

the null-controllability of analytic semigroups: in the smoothness scale of Sobolev spaces defined 

by the generator, if fast controllability holds for initial data in some space, then it holds for 

initial data in any less smooth space; moreover, the same statement holds for fast controllability 

at exponential cost. 

We recall the setting of Section 1.1. A is the boundedly invertible generator of a bounded 

analytic semigroup on the Hilbert space X. For any p>0, let Xp denote the Hilbert space 

D((−A) p ) with the norm xp =(−A) p x (which is equivalent to the graph norm) and let 

X−p be the completion of X with respect to the norm x−p =(−A) −p x. There is a duality 

pairing on X such that X and A are their own dual. For this duality pairing, X−p is the dual 

of Xp. 

For any p ∈ R, the control operator B is said admissible in Xp and fast controllability is 

said to hold in Xp if B satisfies (9) and the property (i) of Lemma 1 holds for all positive T , 

respectively, with X and its norm replaced by Xp and its norm. In this case, the admissibility 

constant KT and the cost CT are denoted by Kp,T and Cp,T . 

Proposition 3. For all real numbers p and p ′ such that p ′ p: 

• Admissibility in Xp implies admissibility in Xp ′. Conversely, if B ∈ L(U, Xp ′) and p′ > 

p − 1/2 then B is admissible in Xp. 

• Fast controllability in Xp implies fast controllability in Xp ′. Moreover, if there are positive 

constants β, C1 and C2 such that Cp,T C2 exp(C1/T β ), then there are positive constants 

C ′ 1 and C′ 2 such that Cp ′ ,T C ′ 2 exp(C′ 1 /T β ). 

Proof. Since A −1 ∈ L(X), Xp ⊂ Xp ′ continuously which proves the first statement. 

Since e tA is an analytic semigroup, it satisfies the smoothing property: ∀n >0, ∀m ∈ R, 

Sn := sup t>0 t n A n e tA L(Xm) < ∞.

By duality, Kp,T is also defined by 

Therefore 


∀x ∈ X−p ′, 

T 

0 

Kp,T C 2 L(X −p ′ ,U) S2 p−p ′ 

 

Ce tA x 2 dt Kp,T x 2 −p . 

T 

0 

t 2(p′ −p) dt 

is finite for 2(p ′ − p) > −1. 

By duality, Cp,T is also defined by the observability inequality: 

∀x ∈ X−p, 

Therefore Cp ′ ,2T Sp−p ′Cp,T /T p−p′ . ✷ 

2.3. Boundary controllability 

 

e T A x −p Cp,T 

 

Ce tA L2 . (0,T ;U) 

This section concerns the following boundary control version of the plate equation with square 

root damping (18): 

¨ζ − ρ˙ζ + 2 ζ = 0 onR+ × M, ζ = 0, ζ = χΓ u on R+ × ∂M, 

ζ(0) = ζ0 ∈ H 1 0 (M), ˙ζ(0) = ζ1 ∈ H −1 (M), u ∈ L 2 loc (R+ × M), (20) 

where χΓ denotes the restriction to the boundary followed by the multiplication by the characteristic 

function of an open subset Γ =∅of ∂M. A key ingredient of the control transmutation 

method of Section 2.4 is the so-called “fundamental controlled solution” for (18). It is constructed 

in Corollary 2 of Theorem 3 in this section which applies [23] to estimate the cost of 

fast boundary controls for (20) when M is a (Euclidean) segment. 

We first adapt the abstract duality framework of Section 1.1 to this boundary control system. 

Here A =−, H0 = L2 (M), U = L2 (Γ ) and α = 1/2. It is convenient to use the state space 

X = H1 × H−1 with the duality pairing 

 

(ζ0,ζ1), (z0,z1) =〈ζ1,z0〉0 +〈ζ0,z1〉0 + ρ A α ζ0,A α 

z0 0 . 

With respect to this pairing, X and A are their own dual, X−1 = H−1 × H−3 is the dual of 

X1 = H3 × H1. Multiplying by ζ(T − t) the dual homogeneous equation 

¨z − ρ˙z + 2 z = 0 onR+ × M, z = z = 0 onR+ × ∂M, (21) 

and integrating by parts on (0,T)× M, yields that the control operator B (arising when rewriting 

the second-order system (20) as the first-order system (6) on ξ(t) = (ζ(t), ˙ζ(t))) is the dual 

(with respect to the new pairing) of the Neumann observation operator C ∈ L(X1,U)defined by


C(z0,z1) = χΓ ∂νz0, where ∂ν is a vector field normal to ∂M. As in the proof of Proposition 3, 

the admissibility of B results from the analyticity of e tA and C ∈ L(Xp; U) with p ∈ (1/4, 1/2). 

(N.b. C ∈ L(Xp,U) for p>1/4 since Xp = H2p+1 × H2p−1 and χ∂M∂ν ∈ L(Hs; U) for 

s>3/2.) 

Theorem 3. For all ρ ∈ (0, 2), there are C1 > 0 and C2 > 0 such that for all L>0 and T ∈ 

]0, inf(π/2,L) 2 ], for all ζ0 and ζ1, there is an input function u such that the solution ζ of (20) 

with M = (−L,L) and Γ ={L} satisfies ζ(T)= ˙ζ(T)= 0 and the cost estimate 

T 

0 

u 2 

L2 dt C2 exp C1L 2 /T ζ0 2 

H 1 +ζ1 2 

H −1 

 

. 

Proof. By Lemma 1, it is enough to prove that the solution of (21) for any initial data z(0) ∈ 

H 1 0 (−L,L) and ˙z(0) ∈ H −1 (−L,L) satisfies the observation inequality 

 

z(T ) 2 

H 1 + ˙z(T ) 2 

H −1 C2 exp(C1/T) 

T 

0 

 

∂sz(t, L) 2 ds. 

The scaling (t, x) ↦→ (σ 2 t,σx) reduces the problem to the case L = π. Using the explicit eigenvalues 

and eigenfunctions of on M = (−π,π), this inequality becomes a “window problem” 

for series of complex exponentials which is almost the one-dimensional setting of “vibrational 

control with structural damping” considered in [23, Section 6], indeed simpler because more 

explicit. Therefore [23, Theorem 1] applies and completes the proof of Theorem 3. ✷ 

Corollary 2. For all ρ ∈ (0, 2) there are positive constants Cρ and C ′ ρ such that, ∀L >0, 

∀T ∈ (0, 1], there is a “fundamental controlled solution” k in C0 ([0,T]; H −1 (]−L,L[)) ∩ 

C1 ([0,T]; H −3 (]−L,L[)) satisfying: 

T 

0 

∂ 2 t k − ρ∂2 s ∂tk + ∂ 4 s k = 0 in D′ ]0,T[×]−L,L[ , 

k⌉t=0 = δ, ∂tk⌉t=0 = δ ′ 

and k⌉t=T = ∂tk⌉t=T = 0, 

 

k(t,·) 2 H −1 (]−L,L[) + ∂tk(t,·) 2 H −3 ′ 

dt C (]−L,L[) ρe CρL2 /T 

. 

Proof. The fast controllability in X = H 1 0 (−L,L) × H −1 (−L,L) stated in Theorem 3 implies, 

by Proposition 3, the fast controllability in X−1 = H −1 (−L,L) × H −3 (−L,L) with the same 

form of cost estimate. Since the Dirac mass at the origin δ is in H s (R) for all s


the controllability cost of a system is not changed by taking its tensor product with a unitary 

group). It applies in particular when M is a rectangle or an infinite strip in the plane controlled 

from one side (this controllability problem in a rectangle with other boundary conditions was 

solved in [11] without the cost estimate, which was added later at the end of [23]). N.b. in this 

example, the condition LΓ < ∞ of [5] required in Theorem 2 is not satisfied. 

Corollary 3. Let ˜M denote another smooth complete Riemannian manifold. For all ρ ∈ (0, 2), 

there are C1 > 0 and C2 > 0 such that, for all L>0 and T ∈ (0, 1] for all ζ0 and ζ1, there is 

an input function u such that the solution ζ of (20) with M = (−L,L) × ˜M and Γ ={L}×∂ ˜M 

satisfies ζ(T)= ˙ζ(T)= 0 and the cost estimate: 

T 

0 

u 2 

L2 dt C2 exp C1L 2 /T ζ0 2 

H 1 +ζ1 2 

H −1 

 

. 

Proof. Let (s, y) denote the variable on M = (−L,L) × ˜M. Denoting respectively by s 

and y the Dirichlet Laplacians on the segment (−L,L) and on ˜M, wehave = s + y. 

Since is boundedly invertible, (20) may also be restated as a first-order system on X = 

H −1 (M) × H −1 (M) by setting ξ(t) = (ζ(t), ˙ζ(t)). Then the semigroup generator A of the 

dual homogeneous system (7) becomes: 

 

0 1 

A = R with R = 

, and e 

−1 −ρ 

tA = e tsR tyR tyR tsR 

e = e e . 

The observation operator Cs defined by Csx = χΓ ∂νz = ∂sz⌉s=L commutes with e tyR . We shall 

estimate the cost by the duality in Lemma 1. Fix the initial state x0 ∈ X and T>0. Applying to 

s ↦→ (e TyR x0)(s, y) for fixed y the observability inequality corresponding to Theorem 3 yields, 

with C ′ T := C2 exp(C1L 2 /T): 

 

0 

L 

 

e TsR 

 

TyR 

e x02 

ds C ′ T 

T 

0 

 

0 

L 

 

Cse tsR 

 

TyR 

e x02 

dsdt. 

Integrating this inequality over ˜M yields (the first and last step use Fubini’s theorem and the 

commutation of operators acting separately on s and y, the second step uses that e tyR is a 

contraction): 

 

M 

 

e T A x0 

 

2 dsdy C ′ T 

C ′ T 

T 

0 

T 

0 

L 

0 

L 

0 

 

˜M 

 

˜M 

 

e TyR 

Cse tsR 

 

x02 

dydsdt 

 

e tyR 

Cse tsR 

 

x02 

dydsdt = C ′ T 

This is the observability inequality corresponding to Corollary 3. ✷ 

T 

0 

 

M 

 

Cse tA 

x02 

dsdydt.


2.4. Proof of Theorem 2 

The first part of Theorem 2 is Corollary 1 for α = 1/2. We shall now prove the second part of 

Theorem 2 by the transmutation control method (cf. [17,19]). According to Proposition 3, it is 

sufficient to consider initial data in the space X2 = H6 × H4 which is smoother than the energy 

space claimed in Theorem 2. 

It results from the work of Bardos–Lebeau–Rauch that (n.b. the control time and the time 

variable are denoted by L and s here): 

Theorem 4. [4,5] Let L>LΩ. For all (w0,w1) and (w2,w3) in H 4 (M) ∩ H 1 0 (M) × H 3 (M) ∩ 

H 1 0 (M) there is an input function v ∈ H 3 (]0,L[×M) supported in ]0,L[×Ω such that the solution 

w ∈ 

n∈N Cn ([0,L]; H 4−n (M)) of 

∂ 2 s w − w = v in ]0,L[×M, w = 0 on ]0,L[×∂M, 

with Cauchy data (w, ∂sw) = (w0,w1) at s = 0, satisfies (w, ∂sw) = (w2,w3) at s = L. Moreover, 

the operator SW defined by SW ((w0,w1), (w2,w3)) = v is continuous in the corresponding 

norms. 

Let T ∈ (0, 1] and L>LΩ be fixed from now on. 

Let (ζ0,ζ1) ∈ X2 = H6 × H4 be an initial data for the plate equation (18). Let v± and w± be 

the input function and solution for the wave equation obtained from Theorem 4 with w0 = ζ0, 

w1 =±ζ1 and w2 = w3 = 0. Let w(±s,·) = w±(s, ·) and v(±s,·) = v±(s, ·) for s ∈[0,L]. We 

define w ∈ 

n∈N Cn (R; H 4−n (M)) and v ∈ H 3 (R × M) as the extensions of w and v by zero 

outside [−L,L]×M. They inherit from w± and v± the properties: 

∂ 2 s w − w = v in D′ (R × M), w = 0 onR × ∂M, 

(w,∂sw )⌉s=0 = (ζ0,ζ1) and (w,∂sw )⌉s=±L = (0, 0). (22) 

Let k, Cρ and C ′ ρ be the fundamental controlled solution and corresponding constants given 

by Corollary 2. We define k as the extension of k by zero outside ]0,T[×]−L,L[. It inherits 

from k the following properties: 

∂ 2 t k − ρ∂2 s ∂tk + ∂ 4 s k = 0 inD′ ]0,T[×]−L,L[ , (23) 

k⌉t=0 = δ, ∂tk⌉t=0 = δ ′ 

and k⌉t=T = ∂tk ⌉t=T = 0, 

T 

0 

 

k(t, ·) 2 H −1 (R) + ∂tk(t, ·) 2 H −3 ′ 

dt C (R) ρe CρL2 /T 

. (24) 

The principle of the control transmutation method is to use k as a kernel to transmute 

w and v into a solution ζ and an input function u for (18). Since k ∈ C 0 (R+; H −1 (R)) ∩ 

C 1 (R+; H −3 (R)), w ∈ H 1 (R; H 3 (M)) ∩ H 3 (R; H 1 (M)) and v ∈ H 1 (R; H 2 (M)) ∩ 

H 3 (R; L 2 (M)), the transmutation formulas


 

u(t, x) = 

R 

 

ζ(t,x)= 

R 

k(t, s)w(s, x) ds, 

ρ∂tk(t, s)v(s, x) + k(t, s) ∂ 2 s + v(s, x) ds 

define functions ζ ∈ C 0 (R+; H 3 (M)) ∩ C 1 (R+; H 1 (M)) and u ∈ L 2 (R+ × M).Thisζ satisfies 

the required initial conditions: k ⌉t=0 = δ and w ⌉s=0 = ζ0 imply ζ⌉t=0 = ζ0; ∂tk ⌉t=0 = ∂sδ and 

∂sw ⌉s=0 = ζ1 imply ∂tζ⌉t=0 = ζ1 by integrating by parts. This ζ satisfies the required final conditions: 

k ⌉t=T = ∂tk ⌉t=T = 0 implies ζ⌉t=T = ∂tζ⌉t=T = 0. This ζ satisfies the required boundary 

conditions: w ⌉∂M = 0 implies ζ⌉∂M = 0 and w ⌉∂M = ∂ 2 s w ⌉∂M = 0 implies ζ⌉∂M = 0. The 

input u is supported in [0,T]×Ω since k is supported in [0,T]×(−L,L) and v is supported 

in (−L,L) × Ω. These ζ and u satisfy the plate equation (18): using (22) in the second step, 

integration by parts in the third, and (23) in the fourth, 

¨ζ − ρ˙ζ + 2 ζ 

 

= ∂ 2 t k w − ρ∂tkw + k 2 w 

 

= 

∂ 2 t k w − ρ∂tk ∂ 2 s w − v + k ∂ 2 2 

s ∂s w − v − v 

 

∂2 = t k − ρ∂ 2 s ∂tk + ∂ 4 s k w + ρ∂tk + k ∂ 2 s + v = u = χΩu. 

Finally, the cost estimate 

u 2 

L2 (R×M) C2 exp CρL 2 /T ζ0 2 

H 6 +ζ1 2 

H 4 

 

results from (24), 

v 2 

H 3 (R×M) 2SW 2 ζ0 2 

H 4 2 

+ζ1 (M) H 3 

and 

(M) 

uL2 (R×M) ρ∂tk L2 (R;H −3 (R)) vH 3 (R;L2 (M)) +kL2 (R;H −1 (R)) vH 1 (R;H 2 (M)) . 

Note added in proof 

After our article was accepted, we became aware of a paper to appear in Asymptotic Analysis: 

“Internal null-controllability for a structurally damped beam equation” by Julian Edward and 

Louis Tebou. This paper concerns (18) when M is a segment and focuses on the limit ρ → 0. 

We claim that our Theorem 2 generalizes the main result of this paper (n.b. LΩ < ∞ always 

hold when the dimension of M is one). Since Theorem 1 of this paper says that the cost does 

not depend on ρ


with α = 1/2) implies |λ±|=μ so that |c| does not depend on ρ here, which completes the proof 

of our claim. 

References 

[1] A. Atallah-Baraket, C. Fermanian Kammerer, High frequency analysis of families of solutions to the equation of 

viscoelasticity of Kelvin–Voigt, J. Hyperbolic Differ. Equ. 1 (4) (2004) 789–812. 

[2] G. Avalos, I. Lasiecka, Optimal blowup rates for the minimal energy null control of the strongly damped abstract 

wave equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (3) (2003) 601–616. 

[3] A.V. Balakrishnan, Damping operators in continuum models of flexible structures: Explicit models for proportional 

damping in beam bending with end-bodies, Appl. Math. Optim. 21 (3) (1990) 315–334. 

[4] C. Bardos, G. Lebeau, J. Rauch, Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation 

de problèmes hyperboliques, in: Nonlinear Hyperbolic Equations in Applied Sciences, 1987, Rend. Sem. Mat. 

Univ. Politec. Torino (1988) 11–31 (special issue). 

[5] C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves 

from the boundary, SIAM J. Control Optim. 30 (5) (1992) 1024–1065. 

[6] G. Chen, D.L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. 

Math. 39 (4) (1982) 433–454. 

[7] S.P. Chen, R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. 

Math. 136 (1) (1989) 15–55. 

[8] S.P. Chen, R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: The case 

0



On action of diffeomorphisms of C*-algebras on 

derivations 

Edward Kissin 

Department of Computing, Communications Technology and Mathematics, London Metropolitan University, 

166-220 Holloway Road, London N7 8DB, UK 

Received 9 January 2006; accepted 13 March 2006 


Communicated by J. Cuntz 

Abstract 

In this paper we consider automorphisms of the domains of closed *-derivations of C*-algebras and show 

that they extend to automorphisms of C*-algebras, so we call them diffeomorphisms. The diffeomorphisms 

generate transformations of the sets of closed *-derivations of C*-algebras. In this paper we study the subgroups 

of diffeomorphisms that define “bounded” shifts of derivations and the subgroups of the stabilizers 

of derivations. 


Keywords: Derivations; C*-algebras; Automorphisms; Diffeomorphisms 


Extensive development of non-commutative geometry requires elaborating of the theory of the 

domains of closed *-derivations of C*-algebras whose properties in many respects are analogous 

to the properties of algebras of differentiable functions. In this paper we consider automorphisms 

of the domains of derivations and show that they extend to automorphisms of C*-algebras, so we 

call them diffeomorphisms. The diffeomorphisms generate transformations of the sets of closed 

*-derivations of C*-algebras. In this paper we study the subgroups of diffeomorphisms that define 

“bounded” shifts of derivations and the subgroups of the stabilizers of derivations. 

E-mail address: e.kissin@londonmet.ac.uk. 


doi:10.1016/j.jfa.2006.03.009

610 E. Kissin / Journal of Functional Analysis 236 (2006) 609–629 

Throughout the paper we denote by (A, ·) a C*-algebra. A closed linear map δ from a 

dense *-subalgebra D(δ) of A into A is called a closed *-derivation if 

δ(AB) = Aδ(B) + δ(A)B and δ A ∗ = δ(A) ∗ 

for A,B ∈ D(δ). 

The subalgebra D(δ) is called the domain of δ; δ is bounded if and only if D(δ) = A. 

Let A be a dense *-subalgebra of A. Denote by Der(A) the set of all closed *-derivations 

δ on A with A = D(δ). We call A a domain if Der(A) = ∅. In Section 2 we show that all 

*-automorphisms of a domain A of A extend to *-automorphisms of A. We call these extensions 

diffeomorphisms of A; they form a group that we denote by Dif(A). Each diffeomorphism φ 

defines a transformation Tφ of Der(A): for every δ ∈ Der(A), the derivation Tφ(δ) = φ−1δφ also 

belongs to Der(A).ThemapT : φ → Tφ is an antirepresentation of the group Dif(A) into the set 

of all transformations of Der(A): Tφθ = Tθ Tφ. We denote by Z(δ) the stabilizer of δ: 

Z(δ) = φ ∈ Dif(A): δ = Tφ(δ) 

and by B(δ) the subgroup of Dif(A) of diffeomorphisms that define bounded shifts of δ: 

B(δ) = φ ∈ Dif(A): the derivation Tφ(δ) − δ is bounded on A in · . 

Denote by B(H) the algebra of all bounded operators on a Hilbert space H and by C(H) the 

ideal of all compact operators. In this paper we study the structure of the groups Z(δ) and B(δ), 

for δ ∈ Der(A), when A are domains of C*-subalgebras A of B(H) that contain C(H). 

An operator F on H with the dense domain D(F) implements δ ∈ Der(A) if 

AD(F ) ⊆ D(F) and δ(A)|D(F) = i[F,A]|D(F) = i(FA − AF )|D(F) for all A ∈ A. (1.1) 

Bratteli and Robinson proved in [2] that, if C(H) ⊆ A ⊆ B(H) and A is a domain of A, then each 

δ ∈ Der(A) has a symmetric implementation: a closed symmetric operator S on H that implements 

δ. The operator S can be chosen (see [5, Theorem 27.21]) to be a minimal implementation, 

that is, for each closed operator F that implements δ, 

S + t1|D(S) ⊆ F for some t ∈ C. 

With each closed symmetric operator S on H , we associate a *-subalgebra 

AS = A ∈ B(H): AD(S) ⊆ D(S), A ∗ D(S) ⊆ D(S) and [S,A]|D(S) is bounded (1.2) 

of B(H). It is the domain (see [5]) of a closed *-derivation δS into B(H) defined by 

δS(A) = i[S,A] for A ∈ AS, 

where [S,A] is the closure of [S,A]|D(S) = (SA − AS)|D(S). Furthermore, AS = B(H) if and 

only if S is bounded. If S is unbounded, δS is unbounded and AS is a Hermitian semisimple 

Banach *-algebra with respect to the norm 

AδS =A+ δS(A) for A ∈ AS.

E. Kissin / Journal of Functional Analysis 236 (2006) 609–629 611 

Denote by FS the closure in ·δS of the set of all finite rank operators in AS and set 

JS = A ∈ AS ∩ C(H): δS(A) ∈ C(H) . 

Then (see [5]) FS and JS are domains of C(H) and closed *-ideals of AS. The *-derivations 

δ min 

S = δS|FS and δ max 

S = δS|JS 

of C(H) are closed; they are the minimal and the maximal closed *-derivations of C(H) with 

minimal implementation S. It was proved in [6] that the closure of (JS) 2 in ·δS coincides with 

FS and that JS = FS if S is selfadjoint. 

In Section 3 we establish a link between minimal symmetric implementations of two deriva- 

tions from Der(A). We prove that if S and T are such implementations, then the algebras FS 

and FT coincide and the norms ·δS and ·δT on them are equivalent. It was shown in [7] 

that if these norms are equal then S − t1 =±UTU∗ for some unitary operator U and t ∈ R. 

In Section 3 we consider the general case and obtain some necessary conditions that S and T 

satisfy. 

Denote by US the group of all unitary operators in the algebra AS and set 

ZS = U ∈ US: δS(U) = λU for some λ ∈ C . 

We show in Section 4 that if C(H) ⊆ A ⊆ B(H) and A is a domain of A, then each φ ∈ Dif(A) is 

implemented by a unitary operator Uφ: φ(A) = UφAU ∗ φ for all A ∈ A. Moreover, if δ ∈ Der(A) 

then φ ∈ B(δ) if and only if Uφ ∈ US, and φ ∈ Z(δ) if and only if Uφ ∈ ZS, where S is a minimal 

symmetric implementation of δ. Identifying Dif(A), B(δ) and Z(δ) with the corresponding 

subgroups of unitary operators, we have 

B(δ) = Dif(A) ∩ US and Z(δ) = Dif(A) ∩ ZS. 

Section 5 is devoted to the investigation of the structure of the groups ZS. In Section 6 we 

study the problem of constructing domains of C*-algebras that extend the domains JS.LetAbe a domain of a C*-subalgebra A of B(H) and let C(H) A. Assume that there is a derivation 

in Der(A) implemented by a symmetric operator S. Then A + JS is a dense *-subalgebra of the 

C*-algebra A + C(H) and δ = δS|(A + JS) is a *-derivation of A + C(H). We provide some 

sufficient conditions for δ to be a closed derivation which implies that A + JS is a domain of A + 

C(H). Numerous examples of such domains can be obtained by considering the *-commutant 

CS = Ker δS = A ∈ AS: δS(A) = 0 

of S. It is a W*-algebra and we prove that, for each C*-subalgebra A of CS satisfying some 

simple conditions, the algebra A + JS is a domain of the C*-algebra A + C(H). In particular, 

CS + JS is a domain of the C*-algebra CS + C(H). Finally, we show that, for each symmetric 

operator S, 

B δ min 

S 

= B δ max 

S 

 

= US and Z δ min 

S 

All symmetric operators in this paper are assumed to be closed. 

max 

= Z δ = Z(δ) = ZS where δ = δS|(CS + JS). 

S


2. Extension of automorphisms from subalgebras of C*-algebras 

Let A be a dense *-subalgebra of a unital C*-algebra A. It is called a Q-subalgebra of A if 

1 ∈ A and Sp A (A) = Sp A (A) for all A ∈ A. (2.1) 

If A is a dense *-subalgebra of a non-unital C*-algebra A, consider the unitizations A = A + C1 

of A and A = A + C1 of A. The algebra A is a Q-subalgebra of A if 

Sp A (A) = SpA (A) for all A ∈ A. 

The domains of closed *-derivations of A are Q-subalgebras of A (see [2,5]). 

Proposition 2.1. Let A be a Q-subalgebra of a C ∗ -algebra A and let φ be a ∗ -automorphism 

of A. Then φ=1,soφ extends to a ∗ -automorphism of A. 

Proof. Let A be unital. Since SpA (A) = SpA (φ(A)), forA∈A,wehave 

SpA (A) = SpA (A) = SpA φ(A) = SpA φ(A) . (2.2) 

If A = A ∗ ∈ A then φ(A) ∗ = φ(A ∗ ) = φ(A) and, by (2.2), 

Hence, for B ∈ A, 

A= sup |λ|= sup |λ|= 

λ∈SpA (A) λ∈SpA (φ(A)) 

φ(A) . 

B 2 = B ∗ B = φ B ∗ B = φ(B) ∗ φ(B) = φ(B) 2 . 

For non-unital A, we have the proof by replacing in the above argument A by A and A by A. ✷ 

For a Q-subalgebra A of a C ∗ -algebra A, denote by Der(A) the set of all closed unbounded 

*-derivations δ on A with A = D(δ). We call A a domain if Der(A) = ∅. We call a 

∗ -automorphism φ of A a diffeomorphism, if it preserves a domain A in A and denote by Dif(A) 

the group of all diffeomorphisms of A that preserve A. Proposition 2.1 yields 

Corollary 2.2. φ → φ|A is an isomorphism from Dif(A) onto the set of all ∗ -automorphisms 

of A. 

Any domain A is a Hermitian semisimple Banach *-algebra (see [5]) with respect to each 

norm 

Aδ =A+ δ(A) for A ∈ A, where δ ∈ Der(A). 

For each bounded derivation δb on A, δ + δb ∈ Der(A). Johnson’s uniqueness of norm theorem 

yields 

Proposition 2.3. All norms ·δ, δ ∈ Der(A), on a domain A are equivalent.


Each φ ∈ Dif(A) defines a transformation Tφ of Der(A) by the formula 

Tφ(δ) = δφ = φ −1 δφ|A, for δ ∈ Der(A). 

Then Tφψ = TψTφ, soT : φ → Tφ is an antirepresentation of the group Dif(A) into the set of all 

transformations of Der(A). Denote by Z(δ) the stabilizer of δ in Dif(A): 

Z(δ) = 

φ ∈ Dif(A): δ = δφ 

and by B(δ) the subgroup of Dif(A) of diffeomorphisms which define bounded shifts of δ: 

B(δ) = φ ∈ Dif(A): the derivation δφ − δ is bounded on A in · . 

If ψ ∈ B(δ) then B(δ) = B(δψ). Denote by A ∗ the dual space of A. 

Proposition 2.4. Let δ ∈ Der(A) and φ ∈ Dif(A). If there exists Δ ∈ Der(A) such that, for each 

A ∈ A and F ∈ A∗ , F(Δφn(A)) → F(δ(A)),asn→∞, then φ ∈ Z(δ). 

Proof. Define Fφ−1 by Fφ−1(A) = F(φ−1 (A)), forA∈A. Then Fφ−1 ∈ A∗ ,so,forA∈A, F Δφn+1(A) 

= Fφ−1 Δφn 

φ(A) → Fφ−1 δ φ(A) 

= F φ −1 δ φ(A) = F δφ(A) . 

Since F(Δ φ n+1(A)) → F(δ(A)), we have F(δφ(A)) = F(δ(A)). Thus δφ(A) = δ(A), so 

φ ∈ Z(δ). ✷ 

3. Domains of C*-algebras containing C(H) 

For x,y ∈ H , the rank one operator x ⊗ y on H acts by the formula 

(x ⊗ y)z = (z, x)y for z ∈ H, and x ⊗ y=xy. 

Let F be an operator on H .Foru, v ∈ H , 

(x ⊗ y)(u ⊗ v) = (v, x)(u ⊗ y), (x ⊗ y) ∗ = y ⊗ x, 

x ⊗ λy = λ(x ⊗ y) = λx ⊗ y, 

F(x⊗ y) = x ⊗ Fy, (x ⊗ y)F = F ∗ x ⊗ y, if y ∈ D(F), x ∈ D F ∗ . (3.1) 

For an algebra of operators A, denote by F(A) the subalgebra of all finite rank operators in A. 

Lemma 3.1. Let A be a domain of A and C(H) ⊆ A ⊆ B(H). Forδ ∈ Der(A), let a symmetric 

operator S be its minimal implementation. Then 

(i) the set of all rank one operators in A consists of all y ⊗ x with x,y ∈ D(S); 

(ii) F(A) ={ n i=1 xi ⊗ yi: xi,yi ∈ D(S)}=F(AS).


Proof. First let us show that the set 

Eδ ={x ∈ H : x ⊗ x ∈ A} 

is a dense linear subspace of H and each rank one operator in A has form y ⊗ x, forx,y ∈ Eδ. 

For each x ∈ H , x=1, the rank one projection x ⊗x belongs to A. It follows from [9, Proposition 

3.4.9] that, for every ε>0, there is a projection pε ∈ D(δ) = A such that x ⊗ x − pε

Using it, we obtain as in (3.2) that 


δ(yn ⊗ yn) = i(yn ⊗ Syn − Syn ⊗ yn) → i(y ⊗ Sy − Sy ⊗ y). 

Since δ is a closed derivation, y ⊗y ∈ D(δ) = A. Hence y ∈ Eδ,soEδ = D(S). Part (i) is proved. 

Clearly, all operators n i=1 xi ⊗ yi, with xi, yi ∈ D(S), belong to F(A). Conversely, each 

A ∈ F(A) has form A = n i=1 xi ⊗ yi, where all xi are linearly independent and all yi are 

linearly independent. Since S implements δ and D(S) is dense in H , 

Az = 

n 

(z, xi)yi ∈ D(S) for all z ∈ D(S). 

i=1 

Hence all yi ∈ D(S). As A ∗ = n i=1 yi ⊗ xi ∈ F(A), all xi ∈ D(S). Thus F(A) = 

{ n i=1 xi ⊗ yi: xi,yi ∈ D(S)}. From this and from [6, Lemma 3.1] it follows that F(A) = 

F(AS). ✷ 

For δ ∈ Der(A), denote by F(A,δ)the closure of F(A) in ·δ. Recall that FS is the closure 

of F(AS) in ·δS . 

Corollary 3.2. Let A be a domain in A, C(H) ⊆ A ⊆ B(H). Let δ,σ ∈ Der(A) and let S,T be, 

respectively, their minimal symmetric implementations. Then 

(i) F(A,δ)is an ideal of A isometrically isomorphic to the algebra (FS, ·δS ). 

(ii) The algebras F(A,δ)and F(A,σ)coincide and D(S) = D(T ). 

Proof. Since S implements δ, it follows from (1.1) that A = D(δ) ⊆ AS and δ = δS|D(δ). Hence 

the norms ·δS and ·δ coincide on A, so it follows from Lemma 3.1(ii) that F(A,δ)and FS 

are isometrically isomorphic. As FS is an ideal of AS (see [6]), F(A,δ)is an ideal of A. 

By Proposition 2.3, the norms ·δ and ·σ on A are equivalent, so the algebras 

F(A,δ) and F(A,σ) coincide. Since A = D(δ) = D(σ), we have from Lemma 3.1(i) that 

D(S) = D(T ). ✷ 

Let S,T be minimal symmetric implementations of δ,σ ∈ Der(A). It follows from Proposition 

2.3 and Corollary 3.2 that the algebras FS and FT coincide and the norms ·δS and ·δT 

on them are equivalent. It was shown in [7, Theorem 4.4] that these norms are equal if and only 

if S − t1 =±UTU∗ for some t ∈ R and a unitary operator U. Below we consider the general 

case and obtain some necessary conditions that S and T satisfy. 

Theorem 3.3. Let A be a domain in A, C(H) ⊆ A ⊆ B(H). Let symmetric operators S and T 

be minimal implementations of δ,σ ∈ Der(A), respectively. Then 

(i) S and T are either both selfadjoint or both non-selfadjoint; 

(ii) there exist bounded invertible operators M from (S − i1)D(S) onto (T − i1)D(T ) and N 

from (S + i1)D(S) onto (T + i1)D(T ) such that 

T − i1 =M(S − i1) and T + i1 = N(S + i1); 

(iii) the derivation σ −δ is bounded if and only if T = S +R where R is selfadjoint and bounded.


Proof. It was shown in [6] that the algebra (FS, ·δS ) has a bounded approximate identity if 

and only if S is selfadjoint. By Corollary 3.2, FS = FT and the norms ·δS and ·δT on them 

are equivalent. This yields (i). 

By Corollary 3.2(ii), D(S) = D(T ). Fixx∈D(S) with x=1. By Proposition 2.3, there is 

C>0 such that, for all y ∈ D(S), 

x ⊗ yσ =x ⊗ y+ σ(x⊗ y) Cx ⊗ yδ = Cx ⊗ y+C δ(x ⊗ y) . 

The operators T − i1 and S − i1 implement σ and δ.Asx ⊗ y=xy, we have from (3.3) 

Therefore 

xy+ x ⊗ (T − i1)y − (T + i1)x ⊗ y 

C xy+ x ⊗ (S − i1)y − (S + i1)x ⊗ y . 

 

(T − i1)y = x ⊗ (T − i1)y x ⊗ (T − i1)y − (T + i1)x ⊗ y + (T + i1)x ⊗ y 

C xy+ x ⊗ (S − i1)y − (S + i1)x ⊗ y + (T + i1)x y 

Cy+C 

(S − i1)y + (S + i1)xy+ (T + i1)xy Ky+C (S − i1)y . 

Since S is symmetric, (S − i1)y2 =Sy2 +y2 . Hence 

 

(T − i1)y (K + C) (S − i1)y for y ∈ D(S). (3.4) 

It is well known that (S ± i1)D(S) are closed subspaces of H and Ker(S ± i1) ={0}. Define 

an operator M from (S −i1)D(S) into (T −i1)D(T ) by M(S−i1)y = (T −i1)y,fory∈ D(S). 

By (3.4), M is bounded. Similarly, the operator R from (T − i1)D(T ) into (S − i1)D(S) defined 

by R(T − i1)y = (S − i1)y, fory∈D(T ), is bounded. Hence R = M−1 . 

Similarly, there is a bounded invertible operator N from (S + i1)D(S) on (T + i1)D(T ) such 

that T + i1 = N(S + i1). Part (ii) is proved. 

For R = R∗ ∈ B(H), the *-derivation δR(A) = i[R,A], A ∈ A, is bounded. Hence δ + δR ∈ 

Der(A) and S + R is its minimal implementation. 

Conversely, let σ − δ be bounded. As D(S) = D(T ), the operator R = T − S is symmetric 

on D(S). There is C>0 such that σ(A)−δ(A) CA for all A ∈ A. Hence, for all x,y ∈ 

D(S), we have from (3.3) that x ⊗ y ∈ A and 

 

 

σ(x⊗ y) − δ(x ⊗ y) = i[T − S,x ⊗ y] =x ⊗ Ry − Rx ⊗ y Cx ⊗ y=Cxy. 

Fix x with x=1. Then 

Ry=x ⊗ Ry x ⊗ Ry − Rx ⊗ y+Rx ⊗ y Cxy+Rxy. 

Hence R is bounded on D(S), so it extends to a selfadjoint bounded operator. ✷


4. Diffeomorphisms of C*-algebras containing C(H) 

Each *-automorphism φ of C(H) is implemented by a unitary operator U: φ(B) = UBU ∗ , 

for B ∈ C(H) (see [8]). This is also true for all *-automorphisms φ of C*-subalgebras A of 

B(H) containing C(H). Indeed, for x,y,∈ H ,setR = φ(x ⊗ y). By (3.1), for all z, u ∈ H , 

R ∗ z ⊗ Ru = R(z ⊗ u)R = φ (x ⊗ y)φ −1 (z ⊗ u)(x ⊗ y) = φ −1 (z ⊗ u)y, x R. 

Hence R is a rank one operator, so φ and φ −1 map finite rank operators into finite rank operators. 

Thus φ(C(H)) = C(H) and there is a unitary U such that φ(B) = UBU ∗ ,forB ∈ C(H).For 

A ∈ A and all x,y ∈ H , 

Ux ⊗ UAy = U(x⊗ Ay)U ∗ = φ A(x ⊗ y) = φ(A)φ(x⊗ y) 

= φ(A)U(x⊗y)U ∗ = φ(A)(Ux ⊗ Uy) = Ux ⊗ φ(A)Uy. 

Hence φ(A)Uy = UAy for all y ∈ H ,soφ(A)= UAU∗ for all A ∈ A. 

Recall that we denote by US the group of all unitary operators in the algebra AS: 

US = U ∈ B(H): U is unitary, UD(S)= D(S) and [S,U]|D(S) is bounded 

and set 

ZS = U ∈ US: δS(U) = λU for some λ ∈ C . 

Theorem 4.1. Let A be a domain in A, C(H) ⊆ A ⊆ B(H), and let φ ∈ Dif(A). Let, as above, 

a unitary U ∈ B(H) implements φ: φ(A)= UAU ∗ for all A ∈ A. Then 

(i) if a symmetric operator S is a minimal implementation of δ ∈ Der(A), then UD(S)= D(S) 

and U ∗ SU is a minimal implementation of the ∗ -derivation δφ; 

(ii) φ ∈ B(δ) if and only if U ∈ US; 

(iii) φ ∈ Z(δ) if and only if U ∈ ZS. 

Proof. By Lemma 3.1, x ⊗ x ∈ A for x ∈ D(S). Hence 

φ(x ⊗ x) = U(x ⊗ x)U ∗ = Ux ⊗ Ux ∈ A, 

so Ux ∈ D(S). Thus UD(S) ⊆ D(S). Since φ −1 ∈ Dif(A) and implemented by U ∗ ,wehave 

U ∗ D(S) ⊆ D(S). Therefore UD(S)= D(S). 

Let T implement δ. For all A ∈ A,wehaveUAU ∗ ∈ A, soUAU ∗ D(T ) ⊆ D(T ). By (1.1), 

δφ(A)|U ∗ D(T ) = φ −1 δ φ(A) U ∗ D(T ) = U ∗ δ UAU ∗ U U ∗ D(T ) 

= U ∗ δ UAU ∗ D(T ) = U ∗ i T,UAU ∗ D(T ) = i U ∗ TU,A U ∗ D(T ) . (4.1) 

Thus U ∗ TU implements δφ. Similarly, if R implements δφ, URU ∗ implements δ. Hence T → 

U ∗ TU is a one-to-one correspondence between the sets of implementations of δ and δφ. Since S 

is a minimal implementation of δ, U ∗ SU is a minimal implementation of δφ. Part (i) is proved.


It follows from (i) and from Theorem 3.3(iii) that δφ − δ is a bounded derivation if and only 

if K = U ∗ SU − S is a bounded operator on D(S). Hence φ ∈ B(δ), if and only if the operator 

[S,U]=UK is bounded on D(S). Since UD(S)= D(S), wehavethatφ ∈ B(δ) if and only if 

U ∈ US. Part (ii) is proved. 

Let φ ∈ Z(δ). Then U ∈ US and, by (i), U ∗ SU is a minimal implementation of δφ and 

D(U ∗ SU) = D(S). Since δφ = δ and S is a minimal implementation of δ, there is λ ∈ C 

such that U ∗ SU = S + λ1|D(S). Hence δS(U) = iλU, soU ∈ ZS. Conversely, if U ∈ ZS then 

U ∗ SU = S + λ1|D(S). AsU ∗ SU is a minimal implementation of δφ, wehaveδφ = δ. ✷ 

Let A be a domain in A, C(H) ⊆ A ⊆ B(H). It follows from Theorem 4.1 that one can identify 

(modulo scalars from the unit circle) the group Dif(A) with the group of all unitary operators 

U on H whose action A → UAU ∗ preserve A. Forδ ∈ Der(A), we will also identify the subgroups 

B(δ) and Z(δ) with the corresponding subgroups of unitary operators. By Theorem 4.1, 

if S is a minimal implementation of δ then 

B(δ) = U ∈ US: UAU ∗ = A = Dif(A) ∩ US, 

Z(δ) = U ∈ ZS: UAU ∗ = A = Dif(A) ∩ ZS. (4.2) 

Proposition 4.2. Let A be a domain in A, C(H) ⊆ A ⊆ B(H), and let S be a minimal symmetric 

implementation of δ ∈ Der(A). Then 

(i) B(δ) is closed in (AS, ·δS ) and Z(δ) is closed in (B(H ), ·); 

(ii) if A is an ideal of AS, then B(δ) = US and Z(δ) = ZS. 

Proof. Let a sequence {Un} of unitaries in B(δ) converge to U in (AS, ·δS ). Then 

U − Un →0 and δS(U) − δS(Un) →0. Hence U is unitary. For each A ∈ A, wehave 

UnAU ∗ n ∈ A, UAU∗ ∈ AS and UAU∗ − UnAU ∗ n →0. Hence 

 

∗ 

δS UAU − δ UnAU ∗ 

n 

= δS(U)AU ∗ + Uδ(A)U ∗ ∗ 

+ UAδS U − δS(Un)AU ∗ n − Unδ(A)U ∗ n 

δS(U)AU ∗ − δS(Un)AU ∗ 

 

n + Uδ(A)U ∗ 

− Unδ(A)U ∗ 

 

n 

+ 

UAδS 

∗ 

U 

− UnAδS → 0. 

U ∗ n 

 

∗ 

− UnAδS U 

n 

Since δ is closed, UAU∗ ∈ A. Thus U ∈ Dif(A).AsU ∈ US, it follows from (4.2) that U ∈ B(δ). 

Let Un ∈ Z(δ), δS(Un) = λnUn, and let U ∈ B(H) and U − Un →0. If λn →∞, then 

Un/λn → 0 and δS(Un/λn) = Un → U. Since δS is a closed derivation, U = 0. This contradiction 

shows that {λn} is bounded. Choose a subsequence converging to some λ and denote it also 

by {λn}. Then δS(Un) = λnUn → λU. Since δS is a closed derivation, U ∈ US and δS(U) = λU. 

Hence Un converge to U in ·δS 

and, as above, U ∈ Dif(A). By (4.2), U ∈ Z(δ). Part (i) is 

proved. 

If A is an ideal of AS then, for each U in US, themapA → UAU ∗ preserves A. Hence 

Dif(A) ⊇ US ⊇ ZS and (ii) follows from (4.2). ✷

5. Structure of the group ZS 


Let U ∈ ZS and δS(U) = λU, forλ∈C. Then U ∗ ∈ US and δS(U ∗ ) = δS(U) ∗ = λU ∗ .As 

∗ ∗ ∗ 

0 = δS(1) = δS U U = U δS(U) + δS U U = (λ + λ)U ∗ U = (λ + λ)1, 

we have Re(λ) = 0. For each t ∈ R,set 

ZS(t) = U ∈ ZS: δS(U) = itU 

and ΓS = t ∈ R: ZS(t) =∅ . 

Then ZS(t)ZS(s) ⊆ ZS(t + s), fort,s ∈ ΓS, soUZS(0) ⊆ ZS(t) and U ∗ ZS(t) ⊆ ZS(0), for 

U ∈ ZS(t). Hence 

ZS(t) = UZS(0) = ZS(0)U for each U ∈ ZS(t), 

ZS(−t)= ZS(t) ∗ , ZS(t + s) = ZS(t)ZS(s) and ZS = 

t∈ΓS 

ZS(t). (5.1) 

All sets ZS(t) are norm closed and ZS(0) is a selfadjoint normal subgroup of the group ZS; ΓS is 

a subgroup of R by addition, isomorphic to the quotient group ZS/ZS(0). 

Denote by Λ(S) and Λ(S ∗ ) the sets of all eigenvalues of operators S and S ∗ and by Hλ(S) 

and Hλ(S ∗ ) the corresponding eigenspaces of S and S ∗ . For a selfadjoint S, letES(λ) be the 

spectral resolution of the identity of S. Then 

ES−t1(λ) = ES(λ + t). (5.2) 

Let t ∈ R −{0}. We say that a unitary operator U on H is an (S,t)-shift if 

Theorem 5.1. 

UD(S)= D(S) and UES(λ)U ∗ = ES(λ + t) for all λ ∈ R. 

(i) If ZS(t) ={0} then the map λ → λ + t is an isomorphism of the sets Sp(S), Λ(S), Sp(S∗ ), 

Λ(S∗ ).ForU∈ ZS(t) and all λ ∈ Λ(S) and μ ∈ Λ(S∗ ), 

∗ 

Hλ+t(S) = UHλ(S) and Hμ+t S ∗ 

= UHμ S . 

(ii) If S is selfadjoint then U ∈ ZS(t), t = 0, if and only if U is an (S, t)-shift. 

Proof. We have UD(S)= D(S), U ∗ D(S) = D(S) and 

Hence 

(SU − US)|D(S) = tU|D(S) and SU ∗ − U ∗ S D(S) =−tU ∗ |D(S). (5.3) 

U ∗ S − (λ + t)1 U D(S) = (S − λ1)|D(S) for each λ ∈ C. 

Therefore λ ∈ Sp(S) if and only if λ + t ∈ Sp(S). Hence λ → λ + t is an isomorphism of Sp(S).


By (5.3), for x ∈ D(S) and y ∈ D(S ∗ ), 

(Sx, Uy) = U ∗ Sx,y = SU ∗ x,y + tU ∗ x,y = x, US ∗ + tU y . 

Hence Uy ∈ D(S ∗ ) and (S ∗ U −US ∗ )y = tUy. Similarly, U ∗ y ∈ D(S ∗ ) and (S ∗ U ∗ −U ∗ S ∗ )y = 

−tU ∗ y. Therefore UD(S ∗ ) = D(S ∗ ), U ∗ D(S ∗ ) = D(S ∗ ) and 

S ∗ U − US ∗ D(S ∗ ) = tU|D(S ∗ ) and S ∗ U ∗ − U ∗ S ∗ D(S ∗ ) =−tU ∗ |D(S ∗ ). (5.4) 

Hence, as above, we have that λ → λ + t is an isomorphism of Sp(S ∗ ). 

For λ ∈ Λ(S), we have from (5.3) that UHλ ⊆ Hλ+t and U ∗ Hλ ⊆ Hλ−t . Therefore 

UHλ ⊆ Hλ+t = UU ∗ Hλ+t ⊆ UHλ. 

Hence UHλ = Hλ+t and λ → λ + t is an isomorphism of Λ(S). Using (5.4), we obtain that the 

same is true for Λ(S ∗ ). Part (i) is proved. 

For any unitary U, the operator USU ∗ is selfadjoint, 

D USU ∗ = UD(S) and EUSU∗(λ) ∗ 

= UES(λ)U 

for all λ ∈ R. (5.5) 

Let U ∈ ZS(t). By (5.3), UD(S) = U ∗ D(S) = D(S) and USU ∗ |D(S) = S − t1| D(S). Hence it 

follows from (5.2) that 

EUSU ∗(λ) = ES−t1(λ) = ES(λ + t) for all λ ∈ R. 

Taking into account (5.5), we have UES(λ)U ∗ = ES(λ + t). Thus U is an (S,t)-shift. 

Conversely, let U be an (S,t)-shift. Then UD(S) = D(S) and UES(λ)U ∗ = ES(λ + t) for 

all λ ∈ R. Hence it follows from (5.2) and (5.5) that 

so USU ∗ |D(S) = S − t1|D(S). Thus 

Therefore U ∈ ZS(t). ✷ 

EUSU ∗(λ) = UES(λ)U ∗ = ES−t1(λ), 

δS(U)|D(S) = i(SU − US)|D(S) = itU|D(S). 

Theorem 5.1 has an especially simple form when S is diagonal, that is, H = 

λ∈Λ(S) Hλ. 

Corollary 5.2. Let S be a diagonal selfadjoint operator. Then t ∈ ΓS if and only if λ → λ + t is 

an isomorphism of Λ(S) and dim Hλ = dim Hλ+t for all λ ∈ Λ(S). 

From Corollary 5.2 it follows that, for any subgroup Γ of R, there is a diagonal S with 

ΓS = Γ . 

If for each t ∈ ΓS, there is Ut ∈ ZS(t) such that U ={Ut: t ∈ ΓS} is a group, then U is called 

a resolving subgroup of ZS. It is commutative and consists of unitary operators Ut , t ∈ ΓS, 

satisfying 

UtD(S) = D(S) and (SUt − UtS)|D(S) = tUt|D(S).


This relation is called the infinitesimal Weyl relation for the group U and the operator S (see [4]). 

It follows from (5.1) that ZS is the semi-direct product of U and the normal subgroup ZS(0). 

Proposition 5.3. If ΓS has a minimal positive element μ, then ΓS ={nμ: n ∈ Z} and, for each 

U ∈ ZS(μ), U ={U n : n ∈ Z} is a resolving subgroup of ZS. 

Proof. We only need to show that ΓS ={nμ: n ∈ Z}. If there is λ ∈ ΓS such that λ = nμ, for 

all n ∈ Z, then mμ


It was proved in [7] that AS = CS + (AS ∩ C(H)) if all dim(Hi)0. 

For λ ∈ Λ(S), letPλbe the projection on the eigenspace Hλ of S. Ifλ= μ, PλPμ = 0. Let 

A ∈ C(H). Each sequence {xλn }, xλn ∈ Hλn with xλn=1 and distinct λn, weakly converges 

to 0. Hence Axλn→0. This implies that ΛA ={λ∈ Λ(S): APλ 

= 0} is a finite or countable 

set: ΛA ={λi}, and APλi →0asi →∞. Thus the series λi∈ΛA 

Pλi APλi converges in norm, 

so 

ρ : A → 

PλAPλ = 

(5.8) 

λ∈Λ(S) 

λi∈ΛA 

Pλi APλi 

is a map from C(H) into C(H) and ρ=1. Let Q = 1 − 

λ∈Λ(S) Pλ and set 

DS = A ∈ C(H): AQ = QA = 0 and PλA = APλ for all λ ∈ Λ(S) . 

Then DS is a C*-subalgebra of C(H). For each A ∈ C(H), ρ(A) ∈ DS and, for each A ∈ DS, 

Lemma 5.5. 

 

A = Q + 

(i) CS is a W ∗ -algebra and 

λ∈Λ(S) 

Pλ 

 

A = ρ(A), so DS = ρ(A): A ∈ C(H) . 

CS = A ∈ B(H): AD(S) ⊆ D(S), A ∗ D(S) ⊆ D(S), [S,A]|D(S) = 0 

= A ∈ B(H): AD(S) ⊆ D(S), AD S ∗ ⊆ D S ∗ , [S,A]|D(S) = S ∗ ,A 

D(S∗ = 0 ) . 

(ii) All Pλ ∈ CS ∩ C ′ S ,forλ ∈ Λ(S), and DS = CS ∩ C(H). 

Proof. The first equality in (i) follows from (1.2) and (5.6). For A ∈ CS, A ∗ ∈ AS and δS(A ∗ ) = 

δS(A) ∗ = 0. Thus A ∗ ∈ CS, soCS is a *-algebra. Denote by Π the last set in (i). Let A ∈ CS. For 

all x ∈ D(S ∗ ) and y ∈ D(S), 

(Ax, Sy) = x,A ∗ Sy = x,SA ∗ y = AS ∗ x,y . 

Hence AD(S∗ ) ⊆ D(S∗ ) and AS∗ |D(S∗ ) = S∗A|D(S∗ ),so[S∗ ,A]|D(S∗ ) = 0. Thus CS ⊆ Π. 

Conversely, let A ∈ Π. Then, for all x ∈ D(S∗ ) and y ∈ D(S), 

∗ ∗ ∗ ∗ ∗ 

S x,A y = AS x,y = S Ax, y = x,A Sy . 

Hence A ∗ y ∈ D(S ∗∗ ) = D(S), soA ∗ D(S) ⊆ D(S). Thus Π ⊆ CS, soΠ = CS.


Let CS ∋ Aλ → A in the weak operator topology (wot). Then A∗ wot 

λ → A∗ .AsS∗∗ = S,wehave, 

for all x ∈ D(S∗ ) and y ∈ D(S), 

∗ ∗ ∗ ∗ 

S x,Ay = A S x,y = limλ AλS ∗ x,y ∗ ∗ 

= lim S Aλx,y λ 

 

= lim λ 

A ∗ λ x,Sy = A ∗ x,Sy = (x, ASy). 

Hence Ay ∈ D(S ∗∗ ) = D(S) and SAy = ASy. Thus AD(S) ⊆ D(S) and [S,A]|D(S) = 0. Similarly, 

A ∗ D(S) ⊆ D(S). Therefore A ∈ CS, soCS is a W*-algebra. Part (i) is proved. 

Since PλD(S) ⊆ Hλ ⊆ D(S) and 

PλS|D(S) = SPλ|D(S) = λPλ|D(S), (5.9) 

we have δS(Pλ) = 0, so Pλ ∈ CS. LetA ∈ CS. Forx ∈ Hλ, wehaveSAx = ASx = λAx, so 

Ax ∈ Hλ. Hence PλAPλ = APλ. Since CS is a *-algebra, PλA = APλ, soPλ ∈ CS ∩ C ′ S . 

Let A ∈ C(H). For each λ ∈ Λ(S), PλAPλ ∈ C(H) and PλAPλD(S) ⊆ Hλ ⊆ D(S). By (5.9), 

δS(PλAPλ)|D(S) = i(SPλAPλ|D(S) − PλAPλS|D(S)) = 0. 

Hence PλAPλ ∈ CS ∩ C(H).Asρ(A) is the norm limit of sums of the operators PλAPλ, λ ∈ ΛA, 

we have ρ(A) ∈ CS ∩ C(H). Thus DS ⊆ CS ∩ C(H). 

Conversely, let A = A ∗ ∈ CS ∩ C(H). Then A = 

i αiPi, where Pi are finite-dimensional 

mutually orthogonal projections from CS ∩ C(H) and |αi|→0. Each subspace PiH lies in D(S) 

and the operator S|PiH is selfadjoint. Hence PiH = 

j Kij , where S|Kij = λij PKij . Therefore 

λij ∈ Λ(S). AsPλ∈ C ′ S , each Pi commutes with all Pλ, soPKij = Pλij PiPλij ∈ DS. Hence 

Pi = 

j PKij ∈ DS.AsDS is norm closed, A ∈ DS. Thus DS = CS ∩ C(H). ✷ 

Recall that a closed subspace L of H (the projection Q on L) reduces a symmetric operator 

S if 

QD(S) ⊆ D(S) and SQ|D(S) = QS|D(S). (5.10) 

The operator S is called simple if it has no reducing subspaces where it induces a selfadjoint 

operator; it is called irreducible if it has no reducing subspaces. 

Denote by H (n) ,1 n ∞, the orthogonal sum of n copies of H and by S (n) the orthogonal 

sum of n copies of S. Lemma 5.5(i) and (5.10) yield 

Lemma 5.6. 

(i) A projection Q reduces a symmetric operator S if and only Q ∈ CS. 

(ii) S is irreducible if and only if CS = C1. 

(iii) If S is irreducible, C S (n) consists of all block-matrix bounded operators (λij 1H ) on H (n) 

with λij ∈ C. 

Let S ∗ be the adjoint of S. The deficiency subspaces N±(S) ={x ∈ D(S ∗ ): S ∗ x =±ix} of S 

areclosedinH and n±(S) = dim N±(S) are called the deficiency indices of S. The operator S 

is selfadjoint if n−(S) = n+(S) = 0; it is maximal symmetric if either n−(S) = 0orn+(S) = 0.


Recall that symmetric operators R on K and S on H are isomorphic if 

UD(R)= D(S) and UR|D(R) = SU|D(R), (5.11) 

for some unitary operator U from K on H .IfS and R are isomorphic, ZS = ZR and ΓS = ΓR. 

The operator T = i d dt on H = L2(0, ∞) with 

D(T ) = h ∈ H : h are absolutely continuous, h ′ ∈ H and h(0) = 0 

is simple and maximal symmetric with n−(T ) = 1, n+(T ) = 0. For each r ∈ R, the multiplication 

operator 

Vrh(t) = e −irt h(t) (5.12) 

on H is unitary. It is easy to check that Vr ∈ UT and δT (Vr) = irVr for all r ∈ R,soVr ∈ ZT (r). 

It is well known (see [1]) that each simple maximal symmetric operator S is isomorphic 

either to T (k) 

if n−(S) = k, n+(S) = 0; or to − T (k) 

if n−(S) = 0, n+(S) = k. (5.13) 

Using Lemma 5.6, we have the following description of ZS for simple maximal symmetric operators. 

Theorem 5.7. Let S be a simple maximal symmetric operator satisfying (5.13), forsomek, and 

let Vr, r ∈ R, be the unitary operators defined in (5.12). Then ΓS = R, {V(r) (k) : r ∈ R} is a 

resolving subgroup of ZS and ZS(0) consists of all unitary block-matrix operators (λij 1H ) on 

H (k) with λij ∈ C. 

We consider now the following criteria for a symmetric operator to be irreducible. 

Lemma 5.8. Let S be a symmetric operator. Let {λn} be eigenvalues of S ∗ with one-dimensional 

eigenspaces: Hn = Chn and let the linear span of all hn be dense in H . Suppose that all 

hn /∈ D(S). If there are μn ∈ C such that hn − μnh1 ∈ D(S), for all n, then S is irreducible. 

Proof. Let a projection Q belong to CS. It commutes with S ∗ ,soS ∗ Qhn = QS ∗ hn = λnQhn. 

Since Hn are one-dimensional, Qhn = αnhn, where αn = 0 or 1. Since Q preserves D(S), 

Q(hn − μnh1) = αnhn − μnα1h1 = αn(hn − μnh1) + (αn − α1)μnh1 ∈ D(S). 

Hence (αn −α1)μnh1 ∈ D(S) for all n. Since all μn = 0, all αn = α1. Thus Q is either 1 or 0. ✷ 

We shall now consider an irreducible non-maximal symmetric operator with a resolving subgroup. 

The symmetric operator S = i d dt on H = L2(0, 2π) with 

D(S) = h ∈ H : h is absolutely continuous, h ′ ∈ H and h(0) = h(2π)= 0 

has n−(S) = n+(S) = 1 (see [1]). It is irreducible. Indeed, S ∗ = i d dt and 

D(S ∗ ) ={h ∈ H : h is absolutely continuous and h ′ ∈ H }.


The functions hn(t) = e int , −∞


(i) δS|A is the maximal closed ∗ -derivation of A implemented by S, that is, (6.1) holds; 

(ii) there exists a bounded linear map θ from C(H) onto A ∩ C(H) such that 

and θ commutes with δ max 

S : 

θ(A)= A for all A ∈ A ∩ C(H), (6.2) 

θ(A)∈ JS and δ max 

max 

S θ(A) = θ δS (A) 

for all A ∈ JS. (6.3) 

Then B = A+JS is a domain of A+C(H), δS|B ∈ Der(B) and S is its minimal implementation. 

Proof. Set δ = δS|B. Since S implements δ and FS ⊆ JS ⊆ B, it is easy to see that S is a minimal 

implementation of δ. Thus we only need to prove that δ is closed. 

By (6.2), C(H) = (A ∩ C(H))∔ Ker(θ) and Ker(θ) is a closed subspace of C(H). Therefore 

A + C(H) = A ∔ Ker(θ) (6.4) 

is the direct sum of A and Ker(θ).LetA ∈ JS. Since θ(A)∈ A ∩ C(H), we have from (6.3) that 

θ(A)∈ A ∩ JS and δS 

max 

max 

θ(A) = δS θ(A) = θ δS (A) ∈ A ∩ C(H). (6.5) 

Hence θ(A) ⊕ δS(θ(A)) belongs to A ⊕ A and to G(δS). Since δS|A satisfies (6.1), we have 

θ(A)∈ A. Therefore A = θ(A)+ (A − θ(A)) and A − θ(A)∈ JS ∩ Ker(θ). Thus, by (6.4), 

B = A + JS = A ∔ JS ∩ Ker(θ) . 

Let An ∈ A, Bn ∈ JS ∩ Ker(θ), A,T ∈ A and B,R ∈ Ker(θ), letAn + Bn → A + B ∈ 

A + C(H) and let δ(An + Bn) → T + R. By (6.5), θ(δmax S (Bn)) = δmax S (θ(Bn)) = 0. Hence 

δmax S (Bn) ∈ Ker(θ). Thus 

δ(An + Bn) = δS(An) + δ max 

S (Bn) → T + R, where δS(An) ∈ A and δ max 

S (Bn) ∈ Ker(θ). 

If a sequence in the direct sum of closed subspaces converges, the components of its elements 

also converge. Hence, by (6.4), An → A, Bn → B, δS(An) → T and δmax S (Bn) → R. As 

δS|A is a closed derivation, we have A ∈ A and δS(A) = T .Asδmax S is a closed derivation, 

B ∈ JS ∩ Ker(θ) and δmax S (B) = R. Thus A + B ∈ B and δ(A + B) = T + R, soδ is a closed 

*-derivation. ✷ 

Corollary 6.2. Let A be a domain of A ⊆ B(H) and δS|A ∈ Der(A) satisfy (6.1). If 

A ∩ C(H) ={0}, then B = A + JS is a domain of A + C(H), δS|B ∈ Der(B) and S is its 

minimal implementation. 

Let A be a C*-subalgebra of CS. Then δS|A = 0 and it satisfies (6.1). For λ ∈ Λ(S), letPλbe the projection on the eigenspace Hλ of S. By Lemma 5.5(ii), PλCSPλ ⊆ CS. Assume also that 

 

Pλ A ∩ C(H) Pλ ⊂ A for all λ ∈ Λ(S). (6.6)


Then all Pλ(A ∩ C(H ))Pλ are C*-algebras. Since δS(A) = 0, for A ∈ CS ∩ C(H), 

CS ∩ C(H) ⊆ CS ∩ JS. (6.7) 

Let A ∈ C(H). Recall (see (5.8)) that APλ = 0 for a finite or countable subset ΛA ={λi} of 

Λ(S), 

APλi →0 and ρ(A) = 

 

PλAPλ = 

λ∈Λ(S) 

 

Pλi 

λi∈ΛA 

APλi ∈ CS ∩ C(H) ⊆ CS ∩ JS, 

where the series converges in norm. 

We will now construct a bounded linear map θ from C(H) onto A∩C(H) satisfying (6.2) and 

(6.3). We will use for this the well-known result that, for each C ∗ -subalgebra B of C(H), there is 

a conditional expectation from C(H) onto B. Thus, for each λ ∈ Λ(S), there is a conditional expectation 

θλ from the algebra PλC(H)Pλ which is isomorphic to C(Hλ) onto the C*-subalgebra 

Pλ(A ∩ C(H ))Pλ of PλC(H)Pλ. Set 

θ(A)= 

λ∈Λ(S) 

θλ(PλAPλ) = 

λi∈ΛA 

θλi (Pλi APλi ) for all A ∈ C(H). (6.8) 

Since θλi (Pλi APλi ) APλi →0 and since θλi (Pλi APλi ) belong to Pλi C(H)Pλi and, 

hence, mutually orthogonal, the series in (6.8) is norm convergent. Hence we have from (6.7) 

that 

θ(A)∈ A ∩ C(H) ⊆ CS ∩ C(H) ⊆ CS ∩ JS for all A ∈ C(H), (6.9) 

so θ maps C(H) into A ∩ C(H). Moreover, θ is linear and bounded, since 

 

θ(A) = supθλi (Pλi APλi ) sup Pλi APλi A. 

Since projections Pλ, λ ∈ Λ(S), are mutually orthogonal, Pλρ(A)Pλ = PλAPλ (see (5.8)). 

Hence 

θ ρ(A) = 

λ∈Λ(S) 

 

θλ Pλρ(A)Pλ = 

λi∈ΛA 

θλi (Pλi APλi ) = θ(A). 

Since θλ(PλAPλ) ∈ Pλ(A ∩ C(H ))Pλ,wehavePλθ(A)Pλ = θλ(PλAPλ), so (see (5.8)) 

ρ θ(A) = 

Pλθ(A)Pλ = 

θλi (Pλi APλi ) = θ(A). 

Thus 

λ∈Λ(S) 

λi∈ΛA 

θ ρ(A) = ρ θ(A) = θ(A) for all A ∈ C(H). (6.10)


Let A ∈ A ∩ C(H). Then A ∈ CS ∩ C(H) and we have from Lemma 5.5(ii) that A = ρ(B) 

for some B ∈ C(H). Since PλAPλ ∈ Pλ(A ∩ C(H ))Pλ and θλ are conditional expectations, 

θλ(PλAPλ) = PλAPλ = Pλρ(B)Pλ = PλBPλ. Thus (6.2) holds, since 

θ(A)= 

θλ(PλAPλ) = 

PλBPλ = ρ(A) = A. 

λ∈Λ(S) 

λ∈Λ(S) 

Let now A ∈ JS. Since PλH ⊂ D(S), for all λ ∈ Λ(S), we have from (5.9) 

Hence 

Pλδ max 

S (A)Pλ = PλδS(A)Pλ = Pλi[S,A]Pλ = i(PλSAPλ − PλASPλ) = 0. 

ρ δ max 

S (A) = 

λ∈Λ(S) 

As δ max 

S (A) ∈ C(H), we have from (6.10) that 

Pλδ max 

S (A)Pλ = 0. 

θ δ max 

S (A) = θ ρ δ max 

S (A) = 0. 

By (6.9), θ(C(H))⊆ CS ∩ JS, so that δ max 

S (θ(A)) = 0. Thus 

δ max 

max 

S θ(A) = 0 = θ δS (A) 

Therefore (6.3) holds and Theorem 6.1 yields 

for all A ∈ JS. 

Theorem 6.3. Let a C ∗ -subalgebra A of CS satisfy (6.6). Then B = A + JS is a domain of the 

C ∗ -algebra A + C(H), δS|B is a closed ∗ -derivation of A + C(H) with minimal implementation 

S. 

Finally, we consider derivations δ with Z(δ) = ZS, where S is a minimal implementation of δ. 

Proposition 6.4. 

(i) Let T be a minimal symmetric implementation of δ ∈ Der(FS). Then B(δ) = UT and 

Z(δ) = ZT . 

(ii) B(δmax S ) = US and Z(δmax S ) = ZS. 

(iii) Let δ = δS|(CS + JS). Then Z(δ) = ZS. 

Proof. As δmin S ,δ∈ Der(FS), it follows from Corollary 3.2 that 

FS = F FS,δ min 

S = F(FS,δ)= FT . 

Since FT is an ideal of AT , Proposition 4.2(ii) yields (i). 

As JS is an ideal of AS, Proposition 4.2(ii) also yields (ii). 

Let U ∈ ZS(t) and A ∈ CS. Since ZS(t) ⊆ AS and CS ⊆ AS,wehaveUAU ∗ ∈ AS. Further 

∗ 

δS UAU = δS(U)AU ∗ + UδS(A)U ∗ ∗ 

+ UAδS U = itUAU ∗ + UA −itU ∗ = 0.


Hence UAU ∗ ∈ CS, soUCSU ∗ = CS.AsJS is an ideal of AS, 

UD(δ)U ∗ = U(CS + JS)U ∗ ⊆ D(δ). 

Thus U ∈ Z(δ), soZS ⊆ Z(δ). Since always Z(δ) ⊆ ZS, wehaveZ(δ) = ZS. ✷ 

Let S be a selfadjoint operator on H = ∞ i=−∞ Hi and let S|Hi = λi1Hi with all distinct λi. 

The group ΓS is described in Corollary 5.2 and CS consists of bounded operators commuting 

with all Pλi . By Theorem 6.3 and Proposition 6.4, δ = δS|(CS + JS) is a closed *-derivation of 

the C*-algebra CS + C(H) and Z(δ) = ZS. 


We are very grateful to Victor Shulman for his valuable suggestions about this paper. 

References 

[1] N.I. Ahiezer, I.M. Glazman, The Theory of Linear Operators in Hilbert Spaces, Ungar, New York, 1961. 

[2] O. Bratteli, D.W. Robinson, Unbounded derivations of C*-algebras, Comm. Math. Phys. 42 (1975) 253–268. 

[3] J. Dixmier, Les C*-algebras et leurs representations, Gauthier–Villars, Paris, 1969. 

[4] P.E.T. Jorgensen, P.S. Muhly, Selfadjoint extensions satisfying the Weyl operator commutation relations, J. Anal. 

Math. 37 (1980) 46–99. 

[5] E. Kissin, V.S. Shulman, Representations on Krein Spaces and Derivations of C*-algebras, Addison– 

Wesley/Longman, Harlow, 1997. 

[6] E. Kissin, V.S. Shulman, Differential Banach *-algebras of compact operators associated with symmetric operators, 

J. Funct. Anal. 156 (1998) 1–29. 

[7] E. Kissin, V.S. Shulman, Dual spaces and isomorphisms of some differential Banach *-algebras of operators, Pacific 

J. Math. 190 (1999) 329–360. 

[8] M.A. Naimark, Normed Rings, Nauka, Moscow, 1968. 

[9] S. Sakai, Operator Algebras in Dynamical Systems, Cambridge Univ. Press, Cambridge, 1991.



AnewprimeC ∗ -algebra that is not primitive 

M.J. Crabb 

Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland, UK 

Received 9 January 2006; accepted 28 February 2006 


Abstract 

An explicit prime non-primitive C∗-algebra is constructed. 


Keywords: C ∗ -algebra; Primitivity 

In [7], N. Weaver gives an example of a C ∗ -algebra that is prime but not primitive, solving a 

long-standing problem. Here we give a simpler example. 

An algebra is said to be prime if the product of any two nonzero two-sided ideals is nonzero. 

An algebra is primitive if it has a faithful irreducible representation. In the case of a C ∗ -algebra 

this representation can be assumed to be a ∗-representation on a Hilbert space [6, Corollary 

2.9.6]. A primitive algebra is always prime. J. Dixmier [5] proved that a separable prime 

C ∗ -algebra is primitive. 

Throughout, X is an uncountable set and G the free group on X with identity e. Forw in G 

in reduced form, define w! to be the set of all prefixes of w, including e and w. For example, 

(xyz)! ={e,x,xy,xyz}. Denote by P the set of all elements of G that in reduced form have 

no negative exponents, and put Q = P −1 , the set of elements with no positive exponents. Thus 

P ∩ Q ={e}. Define L by 

L := {p!∪q!: p ∈ P, q ∈ Q}. 

For a = x1x2 ···xn in P , with xi ∈ X, we say that a has degree n and a −1 has degree −n; we 

define the content of a, con(a) := {x1,x2,...,xn}=con(a −1 ). We also define the degree of e to 

E-mail address: mjc@maths.gla.ac.uk. 


doi:10.1016/j.jfa.2006.02.018

M.J. Crabb / Journal of Functional Analysis 236 (2006) 630–633 631 

be 0 and take con(e) := ∅.ForA⊆P ∪ Q, define con(A) := 

g∈A con(g). The cardinal of a set 

S will be denoted by |S| and the symbol ⊃ will denote strict containment. 

Lemma. 

(L1) A ∈ L, g ∈ A ⇒ g −1 A ∈ L. 

(L2) A,B,C ∈ L, g ∈ A, h ∈ B, A ∪ gB ⊆ ghC ⇒ A ∪ gB ∈ L. 

(L3) For A ∈ L, A has at most one element of each degree. 

(L4) A ∈ L, h ∈ P ∪ Q, hA ⊆ A ⇒ h = e. 

Proof. (L1) Suppose that A = (a −1 )!∪b!, where a,b ∈ P . Then we have that A = a −1 (ab)!= 

a −1 c!, where c = ab. Ifg ∈ A then g = a −1 d for some d ∈ c!, sayc = df with f ∈ P . Then 

g −1 A = d −1 c!=d −1 (df )!=(d −1 )!∪f !∈L. 

(L2) We first show that 

A1,A2,A3 ∈ L, A1 ∪ A2 ⊆ A3 ⇒ A1 ∪ A2 ∈ L. (1) 

Suppose that Ai = pi!∪qi! (pi ∈ P , qi ∈ Q, i = 1, 2, 3). Then p1,p2 ∈ p3! and q1,q2 ∈ q3!. 

Clearly, A1 ∪ A2 = p!∪q!, where p is p1 or p2 and q is q1 or q2. 

Now assume that A,B,C ∈ L, g ∈ A, h ∈ B and A ∪ gB ⊆ ghC. Then g −1 A ∪ B ⊆ hC. 

By (L1), g −1 A ∈ L and since e ∈ hC, alsoh −1 ∈ C, hC ∈ L. By(1),g −1 A ∪ B ∈ L. Since 

g −1 ∈ g −1 A ∪ B, (L1) shows that A ∪ gB = g(g −1 A ∪ B) ∈ L. 

(L3), (L4) Clear. ✷ 

Define M := {(A, g): A ∈ L, g∈ A} and H := l 2 (M), a Hilbert space with inner product 〈|〉. 

We identify elements of M with basis unit vectors of l 2 (M). For(A, g) ∈ M, define a linear 

operator Ag on H by the rule that, for (C, k) ∈ M, 

 

(gC,gk) if A ⊆ gC, 

Ag(C, k) = 

0 otherwise. 

If A ⊆ gC here then e ∈ gC,g−1 ∈ C and gC ∈ L, by(L1),whichgives(gC,gk) ∈ M. Itis 

easily verified that Ag is a partial isometry. Write M := {Ag: (A, g) ∈ M}.IfalsoBh∈M then 

for (C, k) ∈ M we have that 

 

(ghC, ghk) if A ∪ gB ⊆ ghC, 

AgBh(C, k) = 

(2) 

0 otherwise. 

If here A ∪ gB ⊆ ghC then, by (L2), A ∪ gB ∈ L and (A ∪ gB)gh ∈ M. IfA ∪ gB ∈ L then 

(2) gives AgBh = (A ∪ gB)gh. IfA ∪ gB /∈ L then always A ∪ gB ghC and AgBh = 0. For 

Ag ∈ M,also(g −1 A) g −1 ∈ M,by(L1).For(C, k) and (D, f ) in M, 

Ag(C, k) = (D, f ) ⇔ A ⊆ gC = D, gk = f 

⇔ g −1 A ⊆ g −1 D = C, g −1 f = k 

⇔ g −1 A 

g −1(D, f ) = (C, k).

632 M.J. Crabb / Journal of Functional Analysis 236 (2006) 630–633 

Since these are partial isometries it follows that (Ag) ∗ = (g −1 A) g −1. In particular, for A ∈ L, 

Ae is a projection. The above gives, for A,B ∈ L, 

 

(A ∪ B)e if A ∪ B ∈ L, 

AeBe = 

0 otherwise. 

It follows that CM ≡ lin(M) is a ∗-algebra of bounded linear operators on H . Denote by C ∗ M 

its closure, a C ∗ -algebra with identity {e}e. 

Theorem. C ∗ M is prime but not primitive. 

Proof. First consider primeness. Let J and K be nonzero two-sided ideals of C ∗ M.Wefirst 

prove that J contains a projection of M. There exists T ∈ J with T 0 and v = (A, g) ∈ M 

with 〈Tv|v〉 > 0. By replacing T by a scalar multiple we can assume that 〈Tv|v〉=1. We have 

that Aev = Ae(A, g) = v. There is a sequence (Tn) in CM such that Tn → T . Put S := AeTAe ∈ 

J and Sn := AeTnAe; then Sn → S. ForBh ∈ M, AeBhAe is 0 or (A ∪ B ∪ hA)h = Ch (say), 

where C ⊇ A and C = A only if hA ⊆ A and so h = e by (L4). We may therefore write Sn = 

αnAe + Rn, where αn ∈ C and Rn is a linear combination of elements Ch ∈ M with C ⊃ A.For 

each such Ch, we have that |C| > |hA|=|A|, C hA and so Chv = Ch(A, g) = 0. Therefore 

Rnv = 0 and Snv = αnAev = αnv. Hence αn =〈αnv|v〉=〈Snv|v〉→〈Sv|v〉=〈AeTAev|v〉= 

〈TAev|Aev〉=〈Tv|v〉=1asn →∞. 

Suppose that A = p!∪q!, where p ∈ P , q ∈ Q. Choose z in X not in con(C) for any Ch which 

appears in the linear sum expressions for the Rn. Define J ∈ L by J = (pz)!∪(qz −1 )!. IfCh 

appears in the sum for Rn then C ⊃ A and so C contains an element of the degree of pz or qz −1 

but which cannot equal pz or qz −1 .By(L3),J ∪ C/∈ L and so JeCh = 0. Hence JeRn = 0 and 

JeSn = αnJeAe = αnJe → Je, since J ∪ A = J .Also,JeSn → JeS. Therefore Je = JeS ∈ J . 

For j in J , (j −1 J)e = (j −1 J) j −1JeJj ∈ J .Takej = qz −1 which gives j −1 J ⊂ P .The 

ideal K contains a projection Ke of M. We likewise choose k ∈ K such that k −1 K ⊂ Q. Then 

(k −1 K)e ∈ K and JKcontains (j −1 J)e(k −1 K)e = (j −1 J ∪k −1 K)e = 0. This shows that C ∗ M 

is prime. 

Now consider primitivity. For m, n = 0, 1, 2, 3,..., 

(p!∪q!)e: p ∈ P, q ∈ Q, p has degree m, q has degree −n 

consists of orthogonal projections ((L3) and (3)). Let φ be a positive linear functional on C ∗ M. 

Then φ is positive only on a countable subset of the above set. Therefore φ is positive only on a 

countable set of Ae, A in L. Choose x in X not in con(A) for any A with φ(Ae)>0. If B ∈ L 

has x ∈ con(B) then φ(Be) = 0. If also h ∈ B then, since Bh = BeBh, the Cauchy–Schwarz 

inequality gives φ(Bh) = 0. Hence φ vanishes on lin{Bh: x ∈ con(B)}, an ideal of CM. Since 

φ is continuous [2, Section 37, Corollary 9], φ vanishes on its closure, an ideal of C ∗ M. 

Suppose that C ∗ M has an irreducible ∗-representation on a complex space V with inner 

product 〈|〉.Letv ∈ V \ 0 and define a positive linear functional φ on C ∗ M by φ(a) =〈av|v〉. 

From above, φ vanishes on a nonzero ideal I. Letb ∈ I \ 0. For all a,c ∈ C ∗ M, 〈bcv|av〉= 

〈a ∗ bcv|v〉 =φ(a ∗ bc) = 0. Since v is cyclic this gives bV = 0, and the representation is not 

faithful. Therefore C ∗ M has no faithful irreducible representation, i.e. is not primitive. ✷ 

(3)

M.J. Crabb / Journal of Functional Analysis 236 (2006) 630–633 633 

Remark. The set M ∪{0} forms an inverse monoid under the multiplication given by 

 

(A ∪ gB)gh if A ∪ gB ∈ L, 

AgBh = 

0 otherwise. 

This is called the McAlister monoid on X and is the Rees quotient of the free inverse monoid on 

X by the ideal generated by all elements xy −1 and x −1 y with x,y ∈ X and x = y [4]. Barnes [1] 

constructs for any inverse semigroup a representation on Hilbert space; this is used here to 

generate C ∗ M. Weaver’s example in [7] is also generated by an inverse semigroup of partial 

isometries. Related results on l 1 -algebras are described in [3,4]. 


I thank Professor W.D. Munn for introducing me to the notion of the McAlister monoid and 

for help with setting out this paper. 

References 

[1] B.A. Barnes, Representations of the l 1 -algebra of an inverse semigroup, Trans. Amer. Math. Soc. 218 (1976) 361– 

396. 

[2] F. Bonsall, J. Duncan, Complete Normed Algebras, Springer-Verlag, Berlin, 1973. 

[3] M.J. Crabb, The l 1 -algebra of a free inverse monoid, Glasgow University, preprint. 

[4] M.J. Crabb, W.D. Munn, The contracted l 1 -algebra of a McAlister monoid, Glasgow University, preprint. 

[5]J.Dixmier,SurlesC ∗ -algèbres, Bull. Soc. Math. France 88 (1960) 95–112. 

[6] J. Dixmier, C ∗ -algèbres et leurs représentations, Gauthier–Villars, Paris, 1969. 

[7] N. Weaver, A prime C ∗ -algebra that is not primitive, J. Funct. Anal. 203 (2003) 356–361.



Quasiregular representations of the infinite-dimensional 

nilpotent group 

Sergio Albeverio a,1 , Alexandre Kosyak b,∗ 

a Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53115 Bonn, Germany 

b Institute of Mathematics, Ukrainian National Academy of Sciences, 3 Tereshchenkivs’ka, Kyiv 01601, Ukraine 

Received 7 February 2006; accepted 13 March 2006 


Abstract 

In the present work an analog of the quasiregular representation which is well known for locally-compact 

groups is constructed for the nilpotent infinite-dimensional group BN 0 and a criterion for its irreducibility is 

presented. This construction uses the infinite tensor product of arbitrary Gaussian measures in the spaces 

Rm with m>1 extending in a rather subtle way previous work of the second author for the infinite tensor 

product of one-dimensional Gaussian measures. 


Keywords: Infinite-dimensional groups; Nilpotent groups; Quasiregular representations; Irreducibility; Infinite tensor 

products; Gaussian measures; Ismagilov conjecture 


1.1. The setting and the main results 

Let (X, B) be a measurable space and let Aut(X) denote the group of all measurable 

automorphisms of the space X. With any measurable action α : G → Aut(X) of a group 

* Corresponding author. Fax: 38044 2352010. 

E-mail addresses: albeverio@uni.bonn.de (S. Albeverio), kosyak01@yahoo.com, kosyak@imath.kiev.ua 

(A. Kosyak). 

1 SFB 611, BIGS, IZKS, Bonn, BiBoS, Bielefeld–Bonn, Germany; CERFIM, Locarno; Accademia di Architettura, 

USI, Mendrisio, Switzerland. 


doi:10.1016/j.jfa.2006.03.013

S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 635 

G on the space X and a G-quasi-invariant measure μ on X one can associate a unitary 

representation π α,μ,X : G → U(L2 (X, μ)), of the group G by the formula (π α,μ,X 

t f )(x) = 

(dμ(αt−1(x))/dμ(x)) 1/2f(αt−1(x)), f ∈ L2 (X, μ). Let us set α(G) ={αt ∈ Aut(X) | t ∈ G}. 

Let α(G) ′ be the centralizer of the subgroup α(G) in Aut(X): α(G) ′ ={g ∈ Aut(X) |{g,αt}= 

gαtg−1α −1 

t = e ∀t ∈ G}. The following conjecture has been discussed in [23–25]. 

Conjecture 1. The representation π α,μ,X : G → U(L 2 (X, μ)) is irreducible if and only if : 

(1) μ g ⊥ μ ∀g ∈ α(G) ′ \{e} (where ⊥ stands for singular), 

(2) the measure μ is G-ergodic. 

We recall that a measure μ is G-ergodic if f(αt(x)) = f(x)∀t ∈ G implies f(x)= const μ 

a.e. for all functions f ∈ L1 (X, μ). 

In this paper we shall prove Conjecture 1 in the case where G is the infinite-dimensional 

nilpotent group G = BN 0 of finite upper-triangular matrices of infinite order with unities on the 

diagonal, the space X = Xm being the set of left cosets Gm \ BN , Gm being suitable subgroups 

of the group BN of all upper-triangular matrices of infinite order with unities on the diagonal, 

and μ an infinite tensor product of Gaussian measures on the spaces Rm with some fixed m>1. 

A more detailed explanation of the concepts used here is given in the following sections. 

1.2. Regular and quasiregular representations of locally compact groups 

Let G be a locally compact group. The right ρ (respectively left λ) regular representation of 

the group G is a particular case of the representation π α,μ,X with the space X = G, the action 

α being the right action α = R (respectively the left action α = L), and the measure μ being the 

right invariant Haar measure on the group G (see, for example, [8,16,17,37]). 

A quasiregular representation of a locally compact group G is also a particular case of the 

representation π α,μ,X (see, for example, [37, p. 27]) with the space X = H \ G, where H is a 

subgroup of the group G, the action α being the right action of the group G on the space X and 

the measure μ being some quasi-invariant measure on the space X (this measure is unique up to 

a scalar multiple). We remark that in [16,17] this representation has also been called geometric 

representation. 

1.3. Analogs of the regular and quasiregular representations of infinite-dimensional groups and 

the Ismagilov conjecture 

In the present article we will consider the approach which deals with analogs for infinitedimensional 

groups of the regular and quasiregular representations of finite-dimensional groups. 

Let G be an infinite-dimensional topological group. To define an analog of the regular representation, 

let us consider some topological group G, containing the initial group G as a dense 

subgroup, i.e. G = G (G being the closure of G). Suppose we have some quasi-invariant measure 

μ on X = G with respect to the right action of the group G, i.e.α = R, Rt(x) = xt −1 .Inthis 

case we shall call the representation π α,μ,G an analog of the regular representation. We shall 

denote this representation by T R,μ , and the Conjecture 1 is reduced to the following Ismagilov 

conjecture.

636 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 

Conjecture 2. (Ismagilov, 1985) The right regular representation T R,μ : G → U(L 2 (G, μ)) is 

irreducible if and only if : 

(1) μ Lt ⊥ μ ∀t ∈ G \{e}, 

(2) the measure μ is G-ergodic. 

Remark 3. In the case of the right regular representation, the group α(G) ′ = R(G) ′ ⊂ Aut(G) 

obviously contains the group L(G), the image of the group G with respect to the left action. 

The work [11] initiated the study of representations of current groups, i.e. groups C(X,U) of 

continuous mappings X ↦→ U, where X is a finite-dimensional Riemannian manifold and U is a 

finite-dimensional Lie group. 

The regular representation of infinite-dimensional groups, in the case of current groups, was 

studied firstly in [1,4,5,14] (see also the book [6]). An analog of the regular representation for an 

arbitrary infinite-dimensional group G, usingaG-quasi-invariant measure on some completion 

G of such a group, is defined in [18,20]. 

For X = S1 , U a compact or non-compact connected Lie group, Wiener measures on the loop 

groups G = C(X,U) were constructed and their quasi-invariance were proved in [1,4–6,28–32]. 

and the measure μ 

Conjecture 2 was formulated by R.S. Ismagilov for the group G = B N 0 

being the product of arbitrary one-dimensional centered Gaussian measures on the group G = 

B N and was proved for this case in [18,19]. 

The first result in this direction was proved in [33]. For the complex infinite-dimensional Borel 

group Bor c,N 

0 and the standard Gaussian measure on its completion Bor c,N the irreducibility of 

the corresponding regular representation was proved there. Here Bor c,N 

0 (respectively Bor c,N )is 

the group of matrices of the form x = exp t + s where t is a diagonal matrix with a finite number 

of nonzero real elements (respectively arbitrary real elements) and s is a finite (respectively 

arbitrary) complex strictly upper-triangular matrix. 

For the product of arbitrary one-dimensional measures on the group B N Conjecture 2 was 

proved in [21] under some technical assumptions on the measure. 

In [20] Conjecture 2 was proved for the groups of the interval and circle diffeomorphisms. For 

the group of the interval diffeomorphisms the Shavgulidze measure [35] was used, the image of 

the classical Wiener measure with respect to some bijection. For the group of circle diffeomorphisms 

the Malliavin measure [30] was used. 

Whether Conjecture 2 holds in the general case is an open problem. 

In [25] it was shown that Conjecture 1 holds for the inductive limit G = SL0(2∞, R) = 

lim 

−→n SL(2n − 1, R), of the special linear groups (simple groups) acting on a strip of length m ∈ N 

in the space of real matrices which are infinite in both directions, the measure μ being a product 

Gaussian measure. 

Let us consider the special case of a G-space, namely the homogeneous space X = H \ G, 

where H is a subgroup of the group G and μ is some quasi-invariant measure on X (if it exists) 

with respect to the right action R of the group G on the homogeneous space H \ G. In this case 

we call the corresponding representation π R,μ,H\G an analog of the quasiregular or geometric 

representation of the group G (see [22]). 

In [2] Conjecture 1 was proved for the solvable infinite-dimensional real Borel group G = 

Bor N 0 acting on G-spaces Xm , m ∈ N, where X m is the set of left cosets Gm \ Bor N , and Gm 

is some subgroups of the group Bor N of all upper-triangular matrices of infinite order with non-


zero elements on the diagonal. The measure μ on Xm is the product of infinitely many onedimensional 

Gaussian measures on R. 

In [23,24] Conjecture 1 was proved for the nilpotent group G = BN 0 and some G-spaces Xm , 

m ∈ N, being the set of left cosets Gm \B N , where Gm are some subgroups of the group BN .Here 

the measure μ on Xm is the infinite product of arbitrary one-dimensional Gaussian measures 

on R. In this case the variables xpq,1 p


Obviously, BN 0 = lim −→n 

B(n,R) is the inductive limit of the group B(n,R) of real uppertriangular 

matrices with units on the principal diagonal 

 

B(n,R) = I + 

 

xkr ∈ R 

1k


Proof. The right action Rt for t ∈ B N 0 

the point x ∈ X m . ✷ 

changes linearly only a finite number of coordinates of 

Now we can define the representation associated with the right action 

in the natural way, i.e. 

T R,μm B : B N 0 → UL 2 X m ,μ m B 

R,μ 

T m B 

t f (x) = dμ m −1 

B Rt (x) dμ m B (x) 1/2 −1 

f Rt (x) . 

Theorem 5. For the measure μm B the following four statements are equivalent: 

(i) the representation T R,μm B is irreducible; 

(ii) (μm B )Lt ⊥ μm B ∀t ∈ B(m,R) \{e}; 

(iii) (μm B )Lexp(tEpq) ⊥ μm B ∀t ∈ R \{0} ∀1 pm), from the fact that the right action Rt for t ∈ BN 0 

finite number of coordinates of the point x ∈ Xm , and that the group Gm 0 = Gm ∩ BN 0 ⊂ Xm acts 

transitively on itself. In fact it is shown that the measure is ergodic with respect to the action of 

the subgroup Gm 0 ⊂ BN 0 . ✷


3. Idea of the proof of irreducibility 

Proof of Theorem 5. The proof of Theorem 5 is organized as follows: 

(i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (i). 

The parts (i) ⇒ (ii) ⇒ (iii) are evident. The part (iii) ⇔ (iv) follows from Lemma 8, which is 

based on the Kakutani criterion [15]. 

The idea of the proof of irreducibility, i.e. the part (iv) ⇒ (i). Let us denote by A m the von 

Neumann algebra generated by the representation T R,μm B 

A m = T R,μm B 

t 

| t ∈ G ′′ . 

We show that (iv) ⇒[(Am ) ′ ⊂ L∞ (Xm ,μm B )]⇒(i). Let the inclusion (Am ) ′ ⊂ L∞ (Xm ,μm B ) 

holds. Using the ergodicity of the measure μm B (Lemma 6) this proves the irreducibility. Indeed 

in this case an operator A ∈ (Am ) ′ should be the operator of multiplication (since (Am ) ′ ⊂ 

L∞ (Xm ,μm B )) by some essentially bounded function a ∈ L∞ (Xm ,μm B ). The commutation relation 

[A,T R,μm B 

t ]=0 ∀t ∈ BN 0 implies a(R−1 t (x)) = a(x) (mod μm B ) ∀t ∈ BN 0 , so by ergodicity of 

the measure μ m B with respect to the right action of the group BN 0 on the space Xm we conclude 

that A = a = const (mod μm B ). This then proves the irreducibility in Theorem 5, i.e. the part 

[(Am ) ′ ⊂ L∞ (Xm ,μm B )]⇒(i). 

The proof of the remaining part, i.e. the implication (iv) ⇒[(Am ) ′ ⊂ L∞ (Xm ,μm B )] is based 

on the fact that the operators of multiplication by independent variables xpq, 1 p m, p


and the series SL pq (μm ) and Σr pq (m) are defined in Lemmas 8 and 15 (see also (18)). This is done 

in Appendices A–C. 

In Appendix A we define the generalization of the characteristic polynomial for matrix C and 

establish some its properties. These properties are used then in Appendices B and C. For a matrix 

C ∈ Mat(k, C) we set 

Gk(λ) = det Ck(λ), where Ck(λ) = C + 

k 

λrErr, λ= (λ1,...,λk) ∈ C k . 

Lemma A. (See Appendix A, Lemma A.7) For a positive definite matrix C ∈ Mat(k, C), λ ∈ R k 

with λr 0, r = 1,...,k, we have 

r=1 

∂ Gk(λ) 

0, 

∂λp Gl(λ) 

where Gl(λ) = M 12...l 

12...l (Ck(λ)) and 1 p l k. 

The proof of Lemma A is based on the following inequality (see Lemma A.6). 

Lemma B. (Hadamard–Ficher’s inequality [12,13], see also [27]) Let C ∈ Mat(m, R) be a positive 

definite matrix and ∅⊆α, β ⊆{1,...,m}. Then 

 

 

det Cα det 

det Cα∪β 

Cα∩β 

det Cβ 

 

 

 

= 

 

 

 

M(α) M(α∩ β) 

 

M(α∪ β) M(β) 0, 

where Cα for α ={α1,...,αs} denotes the matrix which entries lie on the intersection of 

α1,...,αs rows and α1,...,αs columns of the matrix C and M(α) = M α α (C) = det Cα are corresponding 

minors of the matrix C. 

The “best” approximation of xpq by the generators A R,m 

kn is based on the exact computation 

of the matrix elements 

φp(t) = T R,μm B 

t 1, 1 , t = I + 

p 

r=1 

trErn, (tr) p 

r=1 ∈ Rp , 

of the representation T R,μm B and their generalization (see Appendix B, Lemma B.1), and on 

the finding the appropriate combinations of operator functions of the generators A R,m 

kn (see Remark 

13) to approximate the operators of multiplication by xpq. 

Finally the proof of the inequality Σm >CSm, is based on Lemmas A, B and 16 dealing with 

some inequalities involving the generalized characteristic polynomials. Lemma 16 is proved in 

Appendix C.


Remark 7. We shall firstly prove the approximation of xkn in the above sense for the one vector 

1 ∈ L2 (Xm ,μm B ). Secondly, the approximation also holds for some dense set D of analytic 

vectors in the space L2 (Xm ,μm B ) 

 

D = X α = 

 

1km, k


 

 

det C 

= exp − 

(2π) m 1 

exp(tEpq) 

2 

∗ C exp(tEpq)x, x 

dx = dμBpq(t)(x), 

where (Bpq(t)) −1 = Cpq(t) = exp(tEpq) ∗ C exp(tEpq) (we note that det C = det Cpq(t)). 

Hence, using (2) we get 

We shall prove that 

H μ LI+tEpq 

B ,μB = 

det Cpq(t) + C 

2 

det Cpq(t) det C 

det 2 Cpq(t)+C 

2 

1/4 

 

= 


det Cpq(t)+C 

2 

1/2 

. (3) 

= det C + t2 

4 cppA q q(C), (4) 

where A p q (C), 1 p,q m, denote the cofactors of the matrix C corresponding to the row p 

and the column q. Wehave 

hence 


2 


 

and finally, using (3) we get 

where 

H μ mLI+tEpq m 

B ,μB = 

B (n) = 

1r,sm 

= det C + t2 



2 

∞ 

n=q+1 

4 cppA q q(C) 

= 1 + 


t2 

4 cppbqq, 

1/2 

 

= 1 + t2 

4 cppbqq 

−1/2 H μ LI+tEpq 

B (n) 

 

,μB (n) = 

∞ 

n=q+1 

b (n) 

rs Ers and C (n) := B (n) −1 = 

 

1 + t2 

4 c(n) 

1r,sm 

pp b(n) qq 

c (n) 

rs Ers. 

−1/2 

, 

So using the properties of the Hellinger integral for two Gaussian measures we conclude that 

m LI+tEpq m 

μB ⊥ μB ∀t ∈ R \{0} ⇔ 

∞ 

n=q+1 

 

1 + t2 

4 c(n) 

⇔ S L m 

pq μB =∞. 

pp b(n) qq 

−1/2 

= 0


To prove (4) we set Cpq(t) = exp(tEpq) ∗ C exp(tEpq). Wehaveform ∈ N and 1 p 0, k= 1, 2,...,n. 

We use the same result in a slightly different form with bk = 0, k = 1, 2,...,n, 

min 

x∈R n 

n 

k=1 

akx 2 k 

 

 

 

 

 

n 

 

n 

xkbk = 1 = 

k=1 

b 2 k 

ak 

k=1 

−1 

. (5)

The minimum is realized for 


xk = bk 

 

n 

ak 

k=1 

For any subset I ⊂ N let us denote as before by 〈fn | n ∈ I〉 the closure of the linear space 

generated by the set of vectors (fn | n ∈ I) in a Hilbert space H . 

We note that the distance d(fn+1;〈f1,...,fn〉) of the vector fn+1 in H from the hyperplane 

〈f1,...,fn〉 may be calculated in terms of the Gram determinants Γ(f1,f2,...,fk) corresponding 

to the set of vectors f1,f2,...,fk (see [10]): 

d fn+1;〈f1,...,fn〉 = min 

t=(tk)∈R n 

 

 

 

 

fn+1 + 

b 2 k 

ak 

n 

k=1 

−1 

. 

tkfk 

 

 

 

 

 

2 

= Γ(f1,f2,...,fn+1) 

, (6) 

Γ(f1,f2,...,fn) 

where the Gram determinant is defined by Γ(f1,f2,...,fn) = det γ(f1,f2,...,fn) and 

γ(f1,f2,...,fn) =: γn is the Gram matrix 

Lemma 10. We have 

⎛ 

(f1,f1) (f1,f2) ... 

⎞ 

(f1,fn) 

⎜ (f2,f1) 

γ(f1,f2,...,fn) = ⎜ 

⎝ 

(f2,f2) ... 

. .. 

(f2,fn) ⎟ 

⎠ 

(fn,f1) (fn,f2) ... (fn,fn) 

. 

d fn+1;〈f1,...,fn〉 = 

det γn+1 

det γn 

where dn+1 = ((f1,fn+1), (f2,fn+1),...,(fn,fn+1)) ∈ R n . 

Proof. We may write 

 

n 

 

 

 

k=1 

tkfk − fn+1 

 

2 

 

= 

 

n 

k,m=1 

tktm(fk,fk) − 2 

= (fn+1,fn+1) − γ −1 

n dn+1,dn+1 

 

, 

n 

k=1 

= (γnt,t)− 2(t, dn+1) + (fn+1,fn+1), 

tk(fk,fn+1) + (fn+1,fn+1) 

where t = (t1,t2,...,tn) ∈ Rn . Using (58) for An = γn we get 

(γnt,t)− 2(t, dn+1) = γn(t − t0), (t − t0) − γ −1 

n dn+1,dn+1 

 

, 

where t0 = γ −1 

n dn. Hence we get (see (6)) 

min 

t=(tk)∈R n 

 

 

 

 

fn+1 − 

n 

k=1 

tkfk 

 

 

 

 

 

2 

= min 

t=(tk)∈Rn 

(γnt,t)− 2(t, dn+1) + (fn+1,fn+1)


= (fn+1,fn+1) − γ −1 

n dn+1,dn+1 

 

+ min 

t=(tk)∈Rn 

γn(t − t0), (t − t0) 

= (fn+1,fn+1) − γ −1 

n dn+1,dn+1 

 

. ✷ 

Remark 11. In fact a more general result holds. Let us denote by An+1 the real non-necessarily 

symmetric matrix in R n+1 and by An its n × n block after crossing the element in the last column 

and row, by vn+1 = (a1n+1,a2n+1,...,ann+1), hn+1 = (an+11,an+12,...,an+1n) vectors vn+1, 

hn+1 ∈ R n . If det An = 0 then we have 

an+1n+1 − A −1 

n vn+1,hn+1 

det An+1 

= . (7) 

det An 

Proof. It is sufficient to use the identity (Schur–Frobenius decomposition) 

The generators 

 

An 

An+1 = 

v t n+1 

hn+1 an+1n+1 

 

= 

Akn := A R,m 

kn 

 

An 

 

0 Id A−1 0 1 

= d 

dt T R,μm B 

n vt n+1 

hn+1 an+1n+1 

 

 

 

I+tEkn 

t=0 

 

. ✷ 

of the one-parameter groups I + tEkn have the following form (on smooth functions of compact 

support): 

where 

k−1 

Akn = 

r=1 

xrkDrn + Dkn, 1 k m, k < n, Akn = 

m 

r=1 

xrkDrn, m


is the image of the standard Fourier transform F m in the space L 2 (R m ,dx),i.e.Fmn = 

U(C (n) ) −1 F m U(B (n) ), where 

U B (n) 

L 

1/2 

dμB (n)(x) 

= 

dx 

2 (Rm ,μB (n)) 

L 2 (R m ,dx) 

Fmn 

F m 

L2 (Rm ,μC (n)) 

U C (n) 

dμC (n)(x) 

= 

dx 

L 2 (R m ,dx). 

Since the standard Fourier transform F m is defined as follows: 

 

m 1 

F f (y) = √ exp i(y,x)f(x)dx, 

(2π) m 

and, for D = B (n) respectively D = C (n) 

 

dμD(x) 

U(D)= 

dx 

we have finally for Fmn: 

R m 

1/2 

= 

R m 

1 

((2π) m 

exp 

det D) 1/4 

− 1 

4 

D −1 x,x 

, 

(Fmnf )(y) = U C (n)−1 m (n) 

F U B f (y) 

1 

= 

((2π) m det C (n) 

1 

(n) 

exp C 

) 1/4 4 

−1 

y,y 

1 

√ 

(2π) m 

 

× exp i(y,x)f(x) (2π) m det B (n) 

1/4 

exp − 1 

(n) 

B 

4 

−1 

x,x 

dx 

= exp( 1 

 

4 ((C(n) ) −1y,y)) 

(2π) m det C (n) 

Rm 

exp 

i(y,x) − 1 

4 

Using Fourier transform Fm we obtain for Akn = FmAkn(Fm) −1 : 

Akn = i 

where 

k−1 

 

r=1 

xrkyrn + ykn 

B (n) −1 x,x 

f(x)dx. 

1/2 

 

m 

, 1 k m


For a function f : X m → C we set 

 

Mf = 

X m 

f(x)dμ m B (x). 

To approximate the variables xpq, 1 p


4. To make the expression Σr pq (s,α,m) in (11) larger (to apply then the criterium in 

Lemma 12) we chose s (n) ∈ Rr such that 

 

rp 

(n) 

Mξn s 2 

= max 

rp 

Mξn (s) 2 s∈R r 

(which is possible, |Mξ rp 

n (s)| 2 being continuous and bounded). 

5. With the same aim we chose α (n) 

k in such a way that 

 

 

 

 

 

 

 

Aqn − xpqDpn + 

m 

k=1,k=q 

α (n) 

k Akn 

2 

 

 

1 

= min 

 

(tk)∈R m−1 

 

 

 

 


6. The right-hand side of the previous expression is equal (see (6)) to 

where 

Γ(g1,g2,...,g p q ,...,gm) 

Γ(g1,g2,...,gq−1,gq−1,...,gm) , 

m 

k=1,k=q 

tkAkn 

 

2 

 

1 

. 

 

gk := gkn := Akn1, 1 k m, k = q, g p q := g p qn := (Aqn − xpqDpn)1. (12) 

Proof of Lemma 12. If we put rp 

n tnMξn (s (n) ) = 1 we get 

 

 

 

 

 

r 

tn exp s 

 

n 

l=1 

(n) 

l Aln 

 

m 

α 

k=1 

(n) 

k Akn 

 

2 

 

− xpq 1 

 

 

 

 

 

r 

= tn exp s 

 

n 

l=1 

(n) 

l Aln 

 

m 

Aqn − xpqDpn + xpqDpn + α 

k=1,k=q 

(n) 

k Akn 

 

2 

 

− xpq 1 

 

 

 

 

 

r 

= tn xpq Dpn exp s 

 

n 

l=1 

(n) 

l Aln 

 

− Mξ rp 

(n) 

n s 

 

r 

+ exp s (n) 

l Aln 

 

m 

Aqn − xpqDpn + α (n) 

k Akn 

 

2 

 

1 

 

l=1 

k=1,k=q 

= 

t 

n 

2 

n xpq 2 

 

 

 

r 

 

Dpn exp s 

 

l=1 

(n) 

l Aln 

 

− Mξ rp 

(n) 

n s 

 

2 

 

1 

 

 

 

 

+ 

exp 

 

r 

s (n) 

l Aln 

 

m 

Aqn − xpqDpn + α (n) 

k Akn 

 

2 

 

1 

 

= 

n 

t 2 n 

l=1 

 

xpq 2 c (n) 

pp − Mξ rp 

(n) 

n s 2 k=1,k=q


 

 

 

+ 

exp 

 

r 

l=1 

s (n) 

l Aln 

 


m 

k=1,k=q 

α (n) 

k Akn 

where we have used the equality ξ − Mξ 2 =ξ 2 −|Mξ| 2 : 

 

 

 

r 

 

Dpn exp 

 

l=1 

s (n) 

l Aln 

 

− Mξ rp 

(n) 

n s 

 

 

 

1 

 

 

2 

 

1 

, 

 

=Dpn1 2 − rp 

(n) 

Mξn s 2 = c (n) 

pp − rp 

(n) 

Mξn s 2 . 

Definition. We shall say that two series 

n an and 

n bn with positive members are equivalent 

and shall denote this by 

n an ∼ 

n bn if they are convergent or divergent simultaneously. We 

note that if an > 0, bn > 0, n ∈ N, then we have 

an 

∼ an 

. (13) 

min 

t∈R N 

an + bn 

n∈N 

bn 

n∈N 

Using (5) we get, setting b = (Mξ rp 

n (s (n) )) m+1+N 

n=m+1 ∈ RN , N ∈ N, 

∼ 

m+1+N 

 

n=m+1 

m+1+N 

 

n=m+1 

 

r 

tn exp s 

l=1 

(n) 

l Aln 

 

m 

α 

k=1 

(n) 

k Akn 

 

2 

 

 

 

− xpq 1 

(t, b) =−1 

 

|Mξ rp 

n (s (n) )| 2 

c (n) 

pp −|Mξ rp 

n (s (n) )| 2 +(Aqn − xpqDpn + m k=1,k=p α (n) 

2 

k 

Akn)1 2 

−1 

. ✷ 

DuetoRemark9weshallwriteC (respectively Ĉ) instead of C (n) (respectively Ĉ (n) ), where 

⎛ 

Ĉ (n) ⎜ 

= ⎜ 

⎝ 

c (n) 

11 

c (n) 

12 

c (n) 

1m 

⎛ 

C (n) ⎜ 

= ⎜ 

⎝ 

c (n) 

11 

c (n) 

11 

c (n) 

12 

c (n) 

1m c(n) 

2m 

c (n) 

12 ... c (n) 

1m 

c (n) 

22 ... c (n) 

⎞ 

⎟ 

2m ⎟ 

. 

⎟ 

.. ⎠ , 

... c(n) 

mm 

c (n) 

12 ... c (n) 

+ c(n) 

22 ... 

1m 

c (n) 

. .. 

2m 

c (n) 

2m ... c (n) 

11 + c(n) 

22 +···+c(n) 

⎞ 

⎟ 

⎠ 

mm 

. 

Remark 14. To simplify the further computations we assume that the measures μ B (n) for 2 

n m are standard: B (n) = I . Since μ m B = ∞ n=2 μ B (n) this assumption, which only concerns 

finitely many of the μ B (n)’s, does not change the equivalence class of the initial measure μ m B and 

the equivalence class of the corresponding representation T R,μm B .


Using this remark, notation (12) and Fourier transforms we conclude that 

Γ(g1,g2,...,gm) = det Ĉ, i.e. Γ(g1n,g2n,...,gmn) = det Ĉ (n) , (14) 

since (gq,gp) = (Ĉ)pq, 1 p,q m. Indeed for p = q we have 

(gqn,gpn) = (gpn,gqn) = 

(gpn,gpn) = 

 

p−1 

 

 

 

 

r=1 

p−1 

 

r=1 

xrpyrn + ypn 

 

 

 

 

 

q−1 

xrpyrn + ypn, xsqysn + yqn 

2 

p−1 

= 

r=1 

s=1 

(we reinserted here the upper index n in c (n) 

pq for clarity). 

 

xrp 2 yrn 2 +ypn 2 = 

= (ypn,yqn) = c (n) 

pq , 

p 

r=1 

c (n) 

rr = Ĉ (n) 

pp 

In the following we shall need a variant of Lemma 12 using Remark 13 replacing the 

|Mξ rp 

n (s)| by its maximum Ξ rp 

n . Let us set (see (10) for definition of ξ rp 

n (s)) 

 

n = max 

rp 

Mξn (s) 2 . (15) 

Ξ rp 

s∈R r 

Now we see that using s and α as in parts 4 and 5 of Remark 13 we have 

Σ r pq (s,α,m) 

= 

n 

(13) 

∼ 

n 

(15) 

= 

n 

max s (n) ∈R r |Mξ rp 

n (s (n) )| 2 

c (n) 

pp − maxs (n) ∈Rr |Mξ rp 

n (s (n) )| 2 +(Aqn − xpqDpn + m k=1,k=p α (n) 

c (n) 

max s (n) ∈R r |Mξ rp 

n (s (n) )| 2 

pp +(Aqn − xpqDpn + m k=1,k=p α (n) 

c (n) 

pp + 

Remark 9 

= 

n 

Ξ rp 

n 

Γ(g1n,g2n,...,g p qn,...,gmn) 

Γ(g1n,g2n,...,gq−1n,gq+1n,...,gmn) 

k 

Akn)1 2 

Ξ rp Γ(g1,g2,...,gq−1,gq+1,...,gm) 

cppΓ(g1,g2,...,gq−1,gq+1,...,gm) + Γ(g1,g2,...,g p q ,...,gm) 

k 

Akn)1 2 

= Σ r Ξ 

pq (m) := 

n 

rpΓ(g1,g2,...,gq−1,gq+1,...,gm) (14) Ξ 

= 

Γ(g1,g2,...,gm) 

n 

rp 

n A q q(Ĉ (n) ) 

det Ĉ (n) 

. 

For the latter equality we have used the fact that 

cppΓ(g1,g2,...,gq−1,gq+1,...,gm) + Γ g1,g2,...,g p 

q ,...,gm = Γ(g1,g2,...,gm), 

which follows from (26). Indeed it is sufficient to take in (26) C = Ĉ − cppEqq and λq = cpp. 

Then we have


Γ(g1,g2,...,gm) = det Ĉ = det(Ĉ − cppEqq + cppEqq) 

So we have proved the following lemma. 

= det(Ĉ − cppEqq) + cppA q q(Ĉ − cppEqq) 

= Γ g1,g2,...,g p 

q ,...,gm + cppΓ(g1,g2,...,gq−1,gq+1,...,gm). 

Lemma 15. Let 1 r p


(see (40)) and theirs generalizations (see (42)). We cannot calculate explicitly 

 

n = max 

pq 

Mξn (t) 2 , 

Ξ pq 

t∈R p 

but we are able by Lemmas B.1 and B.2 to obtain the estimation Ξ pq 

n >Ψ pq 

n , 

Ψ pq 

n := 

(M 12...p−1p 

12...p−1q (C(n) 

p,q)) 2 exp(−1) 

M 12...p−1 

12...p−1 (C(n) p )M 12...p 

12...p (C(n) p ) + p k=2 

ˆλk(A 

p 

k (C(n) p )) 2 

(see (47) and (48)). The crucial for proving (17) is Lemma 16 dealing with some inequalities 

involving the generalized characteristic polynomials. This lemma is proved in Appendix C. 

We use the notations of Lemma 8 (see Remark 9): 

Let 

S L m 

pq μB = 

∞ 

n=q+1 

c (n) 

pp b(n) 

qq = 

∞ 

n=q+1 

c (n) 

pp A q q(C (n) 

m ) 

det C (n) 

m 

= 

∞ 

n=q+1 

cppA q q(Cm) 

det Cm 

λ = (λk) m k=1 ∈ Rm , ˆλ = (ˆλk) m k=1 , k−1 

ˆλ1 = 0, ˆλk = crr, 2 k m, (19) 

fq = e 

Ĉm = 

1rp


We have replaced the series 

with the equivalent one 

S L m 

pq μB = 

S L m 

pq μB ∼ 

If we use the equality Ĉm = Cm(ˆλ), we get 

Σm := 

 

1rp


Corollary 17. If SL kn (μm B ) =∞for some 1 k


Proof. Probably Lemma A.1 is known in the literature, but since we do not know any precise 

reference, we provide here a direct proof. Obviously Gm(λ) is a polynomial in the variables 

λ = (λ1,λ2,...,λm) ∈ C m of order m. A direct calculation gives us 

hence 

∂r 

 

1 0 ... 0 0 ... 0 

 

 

0 1 ... 0 0 ... 0 

 

 

. .. 

. 

.. 

 

Gm(λ) 

 

 

 

= 0 0 ... 1 0 ... 0 

 

, 

∂λ1∂λ2 ...∂λr 

c1r+1 c2r+1 ... crr+1 cr+1r+1 + λr+1 ... 

 

cr+1m 

 

 

. .. 

. 

.. 

 

 

 

 

c1m c2m ... crm cr+1m ... cmm + 

 

λm 

∂r 

Gm 

 

∂λ1∂λ2 ...∂λr 

λ=0 

= A 12...r 

12...r (C). 

Similarlywehavefor1i1

For m = 3wehave 


G2(λ) = det C2 + λ1A 1 1 (C2) 

2 

+ λ2 A2 (C2) + λ1A 12 

12 (C2) 

= det C2 + λ1A 1 {1} 

1 C2 λ + λ2A 2 {2} 

2 C2 λ . 

G3(λ) = det C3 + λ1A 1 1 (C3) + λ2A 2 2 (C3) + λ3A 3 3 (C3) + λ1λ2A 12 

12 (C3) + λ1λ3A 13 

13 (C3) 

+ λ2λ3A 23 

23 (C3) + λ1λ2λ3A 123 

123 (C3) 

1 

= det C2 + λ1 A1 (C3) + λ2A 12 

12 (C3) + λ3A 13 

13 (C3) + λ2λ3A 123 

123 (C3) 

2 

+ λ2 A2 (C3) + λ3A 23 

23 (C3) + λ3A 3 3 (C3) 

= det C3 + λ1A 1 [1] 

1 C3 λ + λ2A 2 [2] 

2 C3 λ + λ3A 3 [3] 

3 C3 λ , 

G3(λ) = det C3 + λ1A 1 1 (C3) 

2 

+ λ2 A2 (C3) + λ1A 12 

12 (C3) 

1 

+ λ3 A1 (C3) + λ1A 12 

13 (C3) + λ2A 23 

23 (C3) + λ1λ2A 123 

123 (C3) 

= det C3 + λ1A 1 {1} 

1 C3 λ + λ2A 2 {2} 

2 C3 λ + λ3A 3 {3} 

3 C3 λ . 

For m>3 the proof of (28) and (30) is the same. The identity (29) follows from (28) and (31) 

follows from (30). ✷ 

The proof of Lemma 16 is based on Lemmas A.4, A.6 and A.7 concerning the properties of a 

positive matrices. 

Lemma A.4. (Sylvester [10, Chapter II, Section 3]) Let C ∈ Mat(n, R) and 1 p


where M(α) = Mα α (C), A(α) = Aαα (C) and ˆα ={1,...,m}\α. 

More precisely, see [12, p. 573]; [13, Chapter 2.5, Problem 36]. See also [27, Corollary 3.2, 

p. 34]. 

Let us set as before (see (25)) for λ = (λ1,...,λk) ∈ C k and C ∈ Mat(k, C) 

Gk(λ) = det Ck(λ), where Ck(λ) = C + 

In the following lemma we use the notation for λ = (λ1,...,λk) ∈ C k : 

k 

r=1 

λrErr. 

λ ]l[ = (λ1,...,λl−1, 0,λl+1,...,λk), 1 l k, 

and Gl(λ) = M 12...l 

12...l (Ck(λ)), 1 l k.Forα and β such that ∅⊆α ⊆{1, 2,...,l} and ∅⊆β ⊆ 

{l + 1,...,k}, with l


Similarly, we have 

]p[ 

Gl(λ) = Gl λ ]p[ 

+ λpGl λ p (35) ]p[ 

= Gk λ p 

l+1...k ]p[ 

+ λpGk λ l+1...k pl+1...k , 1 p l. 

pl+1...k 

Finally we get (36). Using the following formula: 

′ 

a + bx 

= 

c + dx 

bc − ad 

(c + dx) 2 

we conclude that (36) implies the identity in (37). 

To prove the inequality in (36) we get 

 

Gk(λ 

 

 

]p[ ) p p Gk(λ ]p[ ) 

Gk(λ ]p[ ) pl+1...k 

pl+1...k Gk(λ ]p[ ) l+1...k 

 

 

 

 

l+1...k 

= 

 

 

 

 

A pl+1...k 

 

A 

= 

 

α α (C) Aα∩β 

A α∪β 

α∪β (C) Aβ 

β (C) 

where C = Ck(λ ]p[ ), α ={p} and β ={l + 1,l+ 2,...,k}. ✷ 

A p p(Ck(λ ]p[ )) A∅ ∅ (Ck(λ ]p[ )) 

pl+1...k (Ck(λ ]p[ )) A l+1...k 

l+1...k (Ck(λ ]p[ )) 

α∩β (C) 

 

 

 

 

(34) 

0, 

Appendix B. Calculation of the matrix elements φp(t) for t ∈ R p , their generalizations 

and Ξ pq 

n 

Let us recall (see (10) and (19)) that ˆλk = k−1 

r=1 crr,2 r m, ˆλ1 = 0 and 

To estimate 

 

n = max 

pq 

Mξn (t) 2 , 1 p q m. (39) 

Ξ pq 

 

max 

t∈R p 

t∈Rp Mξ pq 

n (t) 2 

= max 

ξ pq 

n (t)1, 1 2 , 

t∈R p 

where ξ pq 

n (t) = iyqn exp( p 

r=1 tr Arn) we shall find the exact formulas for the matrix elements 

φp(t) = φ (n) 

p (t) = T R,μm B 

exp( p 

r=1 tr Ern) 1, 1 , t = (tr) p 

r=1 ∈ Rp , 1 p m, (40) 

of the restriction of the representation T R,μm B on the commutative subgroup (exp( p r=1 trErn) | 

t ∈ Rp ) Rp of the group BN 0 and theirs generalization defined below. We note that 

exp( p r=1 trErn) = I + p r=1 trErn. 

For 1 p q, p,q ∈ N we get 

ξ pq 

p 

n (t) = iyqn exp 

r=1 

tr Arn 

p 

= iyqn exp i 

r=1 

tr 

r−1 

 

k=1 

xkrykn + yrn 

 

 

 

 

 

; (41) 

we have used the expression Arn = r−1 

k=1 xkrykn + yrn = r k=1 xkrykn (see (9)). We have


T R,μm B 

exp( p r=1 tr 

= exp 

Ern) 

= exp i 

p 

r=1 

tr Arn 

p p 

k=1 

To obtain ξ pp (t) we generalize the function 

r=k 

p 

= exp i 

xkrtr 

 

T R,μm B 

exp( p 

r=1 tr Ern) 

ykn 

r=1 

 

. 

tr 

r 

k=1 

xkrykn 

in the following way. We replace in the latter identity the vectors (tr,...,tr) ∈ R p−k+1 by 

(trk) p 

r=k ∈ Rp−k+1 and denote the result by ξpp(t): 

⎛ 

⎜ 

ξpp(t) = ξpp ⎝ 

t11 

t21 t22 

t31 t32 ... 

tp1 tp2 ... tpp 

⎞ 

p p 

⎟ 

⎠ := exp i 

k=1 

r=k+1 

 

xkrtrk + tkk 

To obtain ξ pq (t) we consider the function ξpq(t; tqq) = ξpp(t) exp(itqqyqn). Wehave 

Finally we have 

ξ pp (t) = ∂ξpp(t) 

∂tpp 

⎛ 

⎜ 

ξpq(t; tqq) = ξpq ⎝ 

 

 

 

:= exp i 

tkr=tk, 1rkp 

t11 

t21 t22 

t31 t32 ... 

tp1 tp2 ... tpp; tqq 

p p 

k=1 

r=k+1 

xkrtrk + tkk 

 

⎞ 

⎟ 

⎠ 

ykn + tqqyqn 

and ξ pq (t) = ∂ξpq(t; 

 

tqq) 

 

∂tqq 

 

. 

 

ykn 

 

. (42) 

tqq=0,tkr=tk, 1rkp 

Let us define φp(t) = ξpp(t)dμ(x,y), φpq(t) = ξpq(t)dμ(x,y), where μ(x,y) = μI (x) ⊗ 

( ∞ n=m+1 μ C (n)(y)) and μI (x) is the standard Gaussian measure in R × R 2 ×···×R m . 

Using definition (39) and the previous equalities we have finally 

Ξ pp 

 

= max 

∂φp(t) 

 

 

t∈R p 

∂tpp 

Ξ pq 

 

= max 

∂φpq(t) 

 

 

t∈R p 

∂tqq 

2 

2 

tkr=tk, 1rkp 

tqq=0,tkr=tk, 1rkp 

, 

. (43) 

Our aim is to estimate Ξ pq . We shall use the notation Ck := C{1,2,...,k} for Mat(m, C) and 1 

k m (see notation Cα for ∅⊆α ⊆{1,...,m} in Lemma B of Section 3). 

.


Lemma B.1. For 1 p q m and φpq(t) = ξpq(t)dμ(x,y) we have 

φpq 

= 

= 

⎛ 

⎜ 

⎝ 

t11 

t21 t22 

t31 t32 t33 

... 

tp1 tp2 tp3 ... tpp; tqq 

 

R (p−1)(p−2) 

2 +p 

exp i 

1 

√ det C1(t) exp 

 

p p 

k=1 

r=k 

⎞ 

⎟ 

⎠ 

xkrtrk 

 

ykn + tqqyqn 

 

dμ(x,y) 

− 1 

 

(CT , T ) − C1(t) 

2 

−1 d,d 

, (44) 

where we set T = (t11,t22,t33,...,tpp; tqq) ∈ R p+1 , C ∈ Mat(p + 1, C) is defined by 

⎛ 

c11 c12 c13 ... c1p c1q 

⎜ c12 c22 c23 ... c2p c2q 

⎜ c13 c23 c33 ... c3p c3q 

C := Cp,q := C{1,2,...,p,q} := ⎜ 

. 

⎜ 

.. 

⎝ 

c1p c2p c3p ... cpp cpq 

c1q c2q c3q ... cpq cqq 

d = d21(t), d31(t), . . . , dp1(t); d32(t), d42(t),...,dp2(t); ...; dpp−1(t) ∈ R (p−1)(p−2) 

2 , 

drs(t) = trses(t), 1 s


where D(t) = diag(t21,...,tp1; t32,...,tp2; t43,...,tp3; ...; tpp−1). We have 

det C1(t) = 1 + 

p 

 

r=1 1i1

hence 


Using the latter inequality we get (see (48)) 

M 12...p−1 12...p 

12...p−1 (Cp)M12...p (Cp) + 

= A p p(Cp)A ∅ ∅ (Cp) + 

A p 

k (Cp) 2


Fourier transform for the Gaussian measure μC in the space Rm is: 

 

1 

√ 

(2π) m 

Rm 

exp i(y,x)dμC(x) = exp − 1 

2 (Cy,y) 

 

, y ∈ R m . 

Let p = 1. Using (55)–(57) we have 

 

φ1(t11) = 

R 

 

exp(it11y1n)dμ(y)= exp − 1 

2 c11t 2 

11 ; 

 

φ1q(t11; tqq) = 

R2 

exp i(t11y1n + tqqyqn)dμ(y)= exp − 1 

c11t 

2 

2 11 + 2c1qt11tqq + cqqt 2 

qq 

 

; 

Mξ 1q 

(t11) = iyqn exp(it11y1n)dμ(y)= 

R 

∂φ1q(t11; 

 

 

tqq) 

 

∂tqq 

=−c1qt11 exp − 

tqq=0 

1 

2 c2 11t2 

11 , 

 

Mξ 1,q (t11) 2 = c 2 1qt2 11 exp−c11t 2 

11 ; 

Ξ 1q 

= maxMξ 

1q (t11) 2 = c2 1q exp(−1) 

= Ψ 1q , 

we have used the obvious result 

t11∈R 

c11 

 

1 

max f(x)= f = 

x∈R a 

1 

, where f(x)= x exp(−ax), a > 0. (59) 

ea 

This proves (47) for (p, q) = (1,q). 

To prove (44) in the general case we note that 

where 

p 

p 

k=1 

r=k+1 

xkrtrk + tkk 

 

ykn + tqqyqn = a(x) + T,y 

R p+1, 

y = (y1n,y2n,...,ypn; yqn), T = (t11,t22,...,tpp; tqq) ∈ R p+1 , 

a(x) = a1(x), a2(x), . . . , ap(x); 0 ∈ R p+1 , ak(x) = 

x = 

1

φpq(t; tqq) = 

Since 

 

= 


R p+1 

exp i 

p p 

k=1 

r=k 

xkrtrk 

 

ykn + tqqyqn 

 

dμ(x,y) 

exp i a(x) + T,y 

dμ(x,y) = exp − 1 

 

C a(x) + T ,a(x)+ T 

2 

 

dμI (x). 

C a(x) + T ,a(x)+ T = Ca(x), a(x) + 2 a(x),CT + (CT , T ), 

we have 

 

φpq(t; tqq) = exp − 1 

 

(CT , T ) 

2 

 

exp − 1 

 

Ca(x), a(x) + 2 a(x),CT 

2 

 

To calculate the latter integral we use (56). Let us introduce the notation 

We show that 

for some 

We have 

where 

Further 

a(x),CT = 

d(t) = drk(t) 

X = (x12; x13,x23; ...; x1p,...; xp−1p) ∈ R (p−1)(p−2) 

2 . 

Ca(x), a(x) + 2 a(x),CT = C(t)X,X + 2 d(t),X 

d(t) ∈ R (p−1)(p−2) 

 

(p − 1)(p − 2) 

2 and C(t) ∈ Mat 

, R . 

2 

p 

ak(x)(CT )k = 

k=1 

= 

1k


Ca(x), a(x) = 

1k,np 

= 

cknak(x)an(x) = 

 

1k


Let e1(t) = c11t11 + c12t22 = 0sot11 =−c12t22/c11. In this case 

Finally 

(CT , T ) = c11t 2 11 + 2c12t11t22 + c22t 2 22 = 

 

c2 12 

− 2 

c11 

c2 

12 

+ c22 t 

c11 

2 22 

 

c12t11 + c22t22 = − c12c12 

 

+ c22 t22 = 

c11 

c11c22 − c2 12 

c11 

M12 

12 

= 

c11 

= M12 

12 

c11 

 

Mξ 22 (t) 2 = Miy2nexp it11 + it22(x12y1n + y2n) 2 

 

= 

∂φ2(t) 

 

 

= 

M 12 

12 

c11 t22 

We have used the inequality 

2 M 

exp − 12 

12 

c11 t2 

22 

1 + c11t 2 22 

M12 

12 

c11 

Hence if we denote t = (t11,t22) ∈ R 2 we have using (43) 

∂t22 

2 

. 

t 2 22 , 

e1(t)=0,t21=t22 

t 2 22 exp 

 

M12 

12 

− + c11 t 

c11 

2 

22 . 

1 + x exp x, x ∈ R. (62) 

Ξ 22 = max 

t∈R2 

22 

Mξ (t) 2 22 (M 

>Ψ := 12 

12 )2 exp(−1) 

c11(M12 12 + c2 . 

11 ) 

This proves (47) for (p, q) = (2, 2). For (2,q),2


 

 

∂φ2q(t; tqq) 

 

∂tqq 

 

tqq=0 

 

= −(c1qt11 + c2qt22 + cqqtqq) + (c11t11 + c12t22 + c1qtqq)c1qt 2 

21 

∂φ2q(t; tqq) 

∂tqq 

 

tqq=0 

1 + c11t 2 21 

 

× exp − 1 

 

(CT , T ) − C1(t) 

2 

−1 d,d 1 

√ 

det C1(t) , 

 

 

= −(c1qt11 + c2qt22) + (c11t11 + c12t22)c1qt 2 

21 

1 + c11t 2 21 

 

× exp − 1 

 

1 

(CT , T ) √ 

2 det C1(t) 

tqq=0 

Let tqq = 0. We chose d(t) = 0sowehavec11t11 + c12t22 = 0 and t11 = −c12t22 . In this case 

c11 

(CT , T ) = c11t 2 11 + 2c12t11t22 + c22t 2 22 = 

 

c2 12 

− 2 

c11 

c2 

12 

+ c22 t 

c11 

2 22 

 

c1qt11 + c2qt22 = − c12c1q 

 

+ c2q 

c11 

t22 = c11c2q − c12c1q 

c11 

Finally, if we denote t = (t11,t22) ∈ R2 ,wehave 

 

2q 

Mξ (t) 2 

= Miyqn exp it11 + it22(x12y1n + y2n) 2 

 

= 

 

= 

M 12 

1q 

c11 t22 

2 exp − M 12 

1 + c11t 2 22 

By (59) we conclude using (43) that 

Ξ 2q 

= maxMξ 

2q (t) 2 

 

max 

 

t∈R 2 

t22∈R 

12 

c11 t2 22 

This proves (47) for (p, q) = (2,q),2 

∂φ2q(t; tqq) 2 

∂tqq 

2 1q 

t22 

c11 

M 12 

 

 

 

tqq=0,e1(t)=0 

. 

M12 

12 

= 

c11 

t 2 22 , 

t22 = M12 

1q 

t22. 

c11 

∂φ2q(t; tqq) 2 

∂tqq 

 

 

 

tqq=0,e1(t)=0 

 

M12 

12 

exp − + c11 t 

c11 

2 

22 . 

(M12 

1q )2 exp(−1) 

c11(M 12 

12 + c2 11 ) = Ψ 2q . 

 

1 

= √ 

det C1(t) exp 

 

− 1 

 

(CT , T ) − C1(t) 

2 

−1 d,d 

, 

T = (t11,t22,t33), d(t) = d21(t), d31(t), d32(t) , 

d21(t) = t21e1(t), d31(t) = t31e1(t), d32(t) = t32e2(t), 

e1(t) = c11t11 + c12t22 + c13t33, e2(t) = c21t11 + c22t22 + c23t33,

hence 

C = C3 = 


c11 c12 c13 

c12 c22 c23 

c13 c23 c33 

⎛ 

C1(t) = I + C(t) = ⎝ 

= diag(t21,t31,t32) 

 

, C(t) (61) 

⎛ 

= ⎝ 

1 + c11t 2 21 c11t21t31 c12t21t32 

c11t21t31 1 + c11t 2 31 c12t31t32 

c11t 2 21 c11t21t31 c12t21t32 

c11t21t31 c11t 2 31 c12t31t32 

c12t21t32 c12t31t32 c22t 2 32 

⎞ 

⎠ 

c12t21t32 c12t31t32 1 + c22t 2 32 

⎛ 

c11 + t 

⎝ 

−2 

21 

c11 c12 

c11 c11 + t −2 

31 

c12 

c12 c12 c22 + t −2 

32 

⎞ 

⎠ , 

⎞ 

⎠ diag(t21,t31,t32). 

We prove the following inequality for an operator C of order n such that I + C>0: 

det(I + C) exp tr C. (63) 

Indeed by Hadamard inequality (see [7] or [13, Section 2.5.4]) we have for positive operator C 

of order n 

det C 

Using the Hadamard inequality and (62) we have for an operator C such that I + C>0 

i=1 

n 

i=1 

cii. 

n 

det(I + C) (1 + cii) (62) n 

 

n 

exp cii = exp 

i=1 

i=1 

cii 

 

= exp(tr C), 

where we denote by tr C the trace of an operator C in the space C n . Using (63) and (61) we 

conclude that 

det I + C(t) tr C(t) = exp 

p−1 

 

k=1 

where α2 k = p r=k+1 t2 rk since by (61) we have 

Using (26) we get 

tr C(t) = 

1k


Finally we have 

det C1(t) = 1 + c11α 2 1 + c22α 2 2 

= t 2 21t2 31t2 

c11 c11 c12 

 

32 c11 c11 c22 

 

c12 c12 c22 

+ 

 

1 

t2 + 

21 

1 

t2 

c11 c22 

 

 

c12 c22 

 

31 

+ 1 

t2 21t2 

1 

c22 + 

31 t2 21t2 + 

32 

1 

t2 31t2 

1 

c11 + 

32 t2 21t2 31t2 

32 

2 

= 1 + c11 t21 + t 2 

31 + c22t 2 2 

32 + M12 12 t21 + t 2 2 

31 t32 . 

For general n we have by analogy (it proves thus (46)) 

n−1 

det C1(t) = 1 + 

 

r=1 1i1


If we denote ek(t) = n r=1 ckrtrr we get 

(CT , T ) = 

1k,rn 

Under conditions (67) we have 

and 

en(t) = 

n 

r=1 

ckrtrrtkk = 

n 

ek(t)tkk, 

k=1 

A 

cnr 

n r (Cn) 

An n (Cn) tnn = M12...n 

(CT , T ) (69) 

= 

n 

k=1 

For n = 3 using (70) and (71) we can calculate 

12...n (Cn) 

M 12...n−1 

tnn, 

12...n−1 (Cn) 

e3(t) = M123 

123 (C3) 

M12 12 (C3) t33, (CT,T)= M123 

123 (C3) 

M12 12 (C3) t2 33 , 

If, in addition, e1(t) = e2(t) = 0, we have (see (66)) 

2 

tr C(t) = c11 t22 + t 2 

33 + c22t 2 33 = 

 

M12 c11 

1 ∂(CT,T) 

= en(t). (69) 

2 ∂tnn 

∂(C1(t) −1d(t), d(t)) 

= 0 (70) 

∂tnn 

ek(t)tkk = en(t)tnn = M12...n 

12...n (Cn) 

M 12...n−1 

12...n−1 (Cn) t2 nn . (71) 

13 (C3) 

M 12 

23 (C3) 

∂(C1(t) −1d(t), d(t)) 

= 0. 

∂t33 

2 

+ 1 + c22 t 2 33 . 

For n = 3wehaveife1(t) = e2(t) = 0, using the values for t22, e3(t) and (CT , T ) 

 

 

 

∂φ3(t) 2 

 

∂t33 

= e2 3 (t) exp(−(CT , T )) (64) 

e 

det C1(t) 

2 3 (t) exp−(CT , T ) − tr C(t) 

 

M123 123 

= 

(C3) 

M12 12 (C3) 

2 t 2 33 exp 

 

−t 2 

M123 123 

33 

(C3) 

M12 12 (C3) + (c11 

 

M12 + c22) + c11 

We get by (59) 

 

M123 max 

t33∈R 

= 

= 

123 (C3) 

M 12 

12 (C3) 

2 t 2 33 exp 

 

−t 2 

M123 123 

33 

(C3) 

M12 12 (C3) + (c11 + c22) + c11 

M 123 

123 (C3) 

M 12 

12 (C3) 

2 exp(−1) 

M 123 

123 (C3) 

M 12 

12 (C3) + (c11 + c22) + c11 

M 12 

13 (C3) 

M 12 

12 (C3) 

2 

13 (C3) 

M 12 

12 (C3) 

 

M12 13 (C3) 

M12 12 (C3) 

2 (M123 123 (C3)) 2 exp(−1) 

M12 12 (C3)M123 123 (C3) + c11(M12 13 (C3)) 2 + (c11 + c22)(M12 12 (C3)) 2 = Ψ 33 . 

Finally we have (see (43)) 

2 

.


Ξ 33 

 

= max 

33 

Mξ (t) 2 

max 

∂φ3(t) 

 

 

t∈R 2 

This proves (47) for (p, q) = (3, 3). 

By analogy we have for general n: 

 

= − 1 

t33∈R 

∂t33 

2 

e1(t)=e2(t)=0 

Ψ 33 . 

+ ∂(C1(t) −1 1 

d(t), d(t)) exp(− 2 [(CT , T ) − (C1(t) 

∂tnn 

−1d(t), d(t))]) 

√ , 


∂φn(t) ∂(CT,T) 

∂tnn 2 ∂tnn 

 

∂φn(t) 

= −en(t) + 

∂tnn 

∂(C1(t) −1 1 

d(t), d(t)) exp(− 2 [(CT , T ) − (C1(t) 

∂tnn 

−1d(t), d(t))]) 

√ . 


When trk = trr, n r k 2, we have by (65) 

tr C(t) = 

1k


where C = Cn,q and T are defined in Lemma B.1. Moreover, the above conditions gives us the 

same solutions (68) as before, hence using the decomposition of the minor M 12...n−1n 

12...n−1q (Cn,q) we 

have 

eq(t) = (Cn,qT)q = 

n 

r=1 

cqrtrr = 

n 

r=1 

Finally we get if e1(t) =···=en−1(t) = 0 and tqq = 0 

Ξ nq 

 

max 

 

tnn∈R 

∂φnq(t; tqq) 2 

∂tqq 

 

 

 

tqq=0 

12...n−1n 

M 

= max 

tnn∈R M 12...n−1 

12...n−1 (Cn) 

= 

12...n−1q (Cn,q) 

A 

cqr 

n r (Cn) 

An n (Cn) tnn = M12...n−1n 

An n (Cn) 

max 

tnn∈R e2 q (t) exp −(CT , T ) − tr C(t) 

2 t 2 nn exp 

 

−t 2 

M12...n 12...n 

nn 

(Cn) 

(M 12...n−1n 

12...n−1q (Cn,q)) 2 exp(−1) 

M 12...n−1 

12...n−1 

12...n−1q (Cn,q)tnn 

nk=2 ˆλk(A 

+ 

(Cn) n k 

(An n (Cn)) 2 

M 12...n−1 

12...n−1 (Cn)M12...n 12...n (Cn) + n ˆλk(A k=2 n k (Cn)) 2 = Ψ nq . ✷ 

Appendix C. Proof of Lemma 16 

. 

(Cn)) 2 

Proof. Firstly, we prove by induction the inequalities I k k 0fork2. Secondly, we show that 

inequality I k k 0 and Lemma A.6 imply the inequality I k m 0formk where (see (24)): 

I k m := fkA k k Cm(ˆλ) − ˆλkA k 

k Cm ˆλ [k] 0, 2 k m. 

We shall show also that I 2 m 

= 0. In the case m = 2wehave 

I 2 2 = f2A 2 2 C2(ˆλ) − ˆλ2A 2 

2 C2 ˆλ [2] = 0 

since f2 = ˆλ2 = c11 by (19), (20) and (49), and 

A 2 2 C2(ˆλ) = A 2 

2 C2 ˆλ [2] = A 2 2 (C2) = c11, 

where 

 

c11 

C2(ˆλ) = 

c12 

c12 c11 + c22 

 

 

, C2ˆλ 

[2] = C2 = 

c11 c12 

c12 c22 

In the case m = 3 we prove the following inequalities: 

I 2 3 := f2A 2 2 C3(ˆλ) − ˆλ2A 2 

2 C3 ˆλ [2] 0, (72) 

I 3 3 := f3A 3 3 C3(ˆλ) − ˆλ3A 3 

3 C3 ˆλ [3] 0. (73) 

Since (see (21)) 

 

.


C3(ˆλ) = 

c11 c12 c13 

c12 c11 + c22 c23 

c13 c23 c11 + c22 + c33 

 

 

, C3ˆλ 

[2] = 

c11 c12 c13 

c12 c22 c23 

c13 c23 c11 + c22 + c33 

and C3(ˆλ [3] ) = C3 we have by (26) 

A 2 2 C3(ˆλ) = A 2 

2 C3 ˆλ [2] = A 2 2 (C3) + ˆλ3A 23 

23 (C3), A 3 

3 C3 ˆλ [3] = A 3 3 (C3). 

The latter equalities give us I 2 3 = 0. This proves (72). Indeed we have 

I 2 3 = 

ˆλ2 

2 

A2 (C3) + ˆλ3A 23 

23 (C3) 

− ˆλ2 

2 

A2 (C3) + ˆλ3A 23 

23 (C3) ≡ 0. 

Since f2 = ˆλ2 = c11 and ˆλ1 = 0wehaveA2 2 (Cm(ˆλ)) = A2 2 (Cm(ˆλ [2] )) hence 

I 2 m := f2A 2 2 Cm(ˆλ) − ˆλ2A 2 

2 Cm ˆλ [2] ≡ 0, 2 m. (74) 

Remark C.1. In what follows we take λ = (λr) k 1 ∈ Rk with λ1 = 0. 

To prove (73) for k = 3 we use the identity for λ = (0,λ2) ∈ R 2 , ˆλ = (0,c11), 

A 3 3 

A 3 3 

 

C3(ˆλ) = M 12 

12 C3(ˆλ) = M 12 

12 (C3) + c 2 11 

 

C3(λ) = M 12 

12 C3(λ) = M 12 

12 (C3) + λ2c11, 

= M12 

12 (C3) + ˆλ2c11, 

∂M 12 

12 (C3(λ)) 

∂λ2 

= c11. 

We have 

I 3 3 := f3A 3 3 C3(ˆλ) − ˆλ3A 3 

3 C3 ˆλ [3] 

 

= c11 + c2 12 (M 

+ 

c11 

12 

12 (C3)) 2 

c11(M12 12 (C3) + c2 11 ) 

 

M12 12 (C3) + c 2 

11 − (c11 + c22)M 12 

12 (C3) 

 

= c11 + c2 12 

+ 

c11 

(M12 

12 (C3)) 2 

c11M 12 

12 (C3(ˆλ)) 

 

M 12 

12 C3(ˆλ) − (c11 + c22)M 12 

12 (C3), 

we use here the definition of fq = e 

1rp


Since I 3 3 = I 3 3 (ˆλ) it is sufficient to prove that I 3 3 (λ) > 0forλ2 > 0. We show that 

I 3 3 (0) = 0 and 

Indeed we have M 12 

12 (C3(0)) = M 12 

12 (C3) hence 

I 3 

3 (0) = c11 + c2 12 

c11 

= M 12 

12 (C3) 

+ M12 

12 (C3) 

c11 

c 2 12 + M 12 

∂I3 3 (λ) 

 

= 

∂λ2 

∂I3 3 (λ) 

> 0. 

∂λ2 

 

M 12 

12 (C3) − (c11 + c22)M 12 

12 (C3) 

12 (C3) 

− c22 

c11 

c11 + c2 12 

c11 

 

= 0 and 

 

c11 > 0. 

Finally I 3 3 (λ) > 0forλ2 > 0soI 3 3 = I 3 3 (ˆλ) = I 3 3 (0,c11)>0 and (73) is proved. To prove that 

I k k 0 let us denote f q = e q−1 

r=1 Ψ rq−1 . Using (20) we have 

fq = e 

1rpI4 4 (λ)| λ=ˆλ where I 4 4 (λ) is defined by the formula 

r=1


I 4 

(c11 + c22)M 

4 (λ) := 

12 

12 (C4) 

where 

M12 12 (C4(λ)) 

+ c2 13 

c11 

+ (M12 

13 (C4)) 2 

c11M 12 

12 

(C4(λ)) + 

M 12 

12 

(M123 123 (C4)) 2 

(C4)M 123 

123 (C4(λ)) 

× M 123 

123 C4(λ) − (c11 + c22 + c33)M 123 

123 (C4) 

 

a2 

= a1 + 

M12 12 (C4(λ)) 

 

M 123 

123 C4(λ) + b1 = a1M 123 

123 C4(λ) M 

+ a2 

123 

123 (C4(λ)) 

M12 + b1, 

12 (C4(λ)) 

a1 = c2 13 

> 0, a2 = (c11 + c22)M 

c11 

12 

12 (C4) + (M12 

13 

(C4)) 2 

c11 

b1 = (M123 

123 (C4)) 2 

M12 12 (C4) − (c11 + c22 + c33)M 123 

123 (C4). 

We prove that I 4 4 (λ) 0forλ = (0,λ2,λ3), when λ2 0, λ3 0. It then gives us I 4 4 

I 4 4 (ˆλ) 0. We have (see below the proof of I k k (0) = 0, k 3) 

I 4 

c2 13 

4 (0) = 

c11 

+ (M12 

13 (C4)) 2 

c11M 12 

12 

12 

> 0, 

M123 

123 

+ 

(C4) (C4) 

M12 

− c33 M 

(C4) 123 

123 (C4) = 0. 

Moreover, by inequality (37) of Lemma A.7 we have for λ2 0, λ3 0 

Let us consider the function 

We have 

∂I4 4 (λ) ∂M 

= a1 

∂λ2 

123 

123 (C4(λ)) ∂ M 

+ a2 

∂λ2 ∂λ2 

123 

123 (C4(λ)) 

M12 0, 

12 (C4(λ)) 

∂I4 4 (λ) 

 

a2 

= a1 + 

∂λ3 M12 12 (C4(λ)) 

 

∂M123 123 (C4(λ)) 

0. 

∂λ3 

i 4 4 (t) = I 4 4 (t ˆλ) = I 4 4 (0,tˆλ2,tˆλ3), t ∈ R. 

i 4 4 (0) = I 4 4 (0) = 0 and 

di4 4 (t) 

dt = ∂I4 4 (λ) 

∂λ2 

hence i4 4 (t) 0 by the previous inequalities for t>0. So 

To prove that I k k (ˆλ) 0 we show that 

I 4 4 >I4 4 (0, ˆλ2, ˆλ3) = i 4 4 (t) t=1 0. 

I k k (0) = 0, 2 k and 

ˆλ2 + ∂I4 4 (λ) 

ˆλ3 0 

∂λ3 

∂Ik k (λ) 

0, 2 p


To define the function I k+1 

k+1 k+1 

k+1 (λ) with Ik+1 Ik+1 (ˆλ) we have 

I k+1 

k+1 

k+1 

= fk+1Ak+1 Ck+1(ˆλ) − ˆλk+1A k+1 

k+1 (Ck+1) 

 

= fk + f k+1 A k+1 

Ck+1(λ) − ˆλk+1A k+1 

(75) 

(76) 

 

(54) 

 

k+1 

ˆλkA kk+1 

kk+1 (Ck+1) 

A kk+1 

kk+1 

(Ck+1(λ)) + e 

ˆλkA kk+1 

kk+1 (Ck+1) 

A kk+1 

kk+1 

:= I k+1 

k+1 (ˆλ), 

(Ck+1(λ)) + e 

k 

r=1 

k 

r=1 

Ψ rk 

Ψ rk 

0 

 

 

A k+1 

k+1 

A k+1 

k+1 

k+1 (Ck+1) λ=ˆλ 

Ck+1(λ) − ˆλk+1A k+1 

Ck+1(λ) − ˆλk+1A k+1 

where the function I k+1 

pq 

k+1 (λ) is defined by (see definition (54) of Ψ0 ): 

r=2 

 

 

 

k+1 (Ck+1) 

 

λ=ˆλ 

 

 

 

k+1 (Ck+1) 

 

λ=ˆλ 

I k+1 

 

ˆλkM 

k+1 (λ) = 

12...k−1 

12...k−1 (Ck+1) 

M 12...k−1 

12...k−1 (Ck+1(λ)) + c2 k (M 1k 

+ 

c11 

r=2 

12...r−1r 

12...r−1k 

(Ck+1)) 

2 

M 12...r−1 

12...r−1 

(Ck+1)M12...r 12...r (Ck+1(λ)) 

 

× M 12...k 

12...k Ck+1(λ) − ˆλk+1M 12...k 

12...k (Ck+1) 

 

ˆλkM 

= 

12...k−1 

12...k−1 (Ck+1) 

M 12...k−1 

12...k−1 (Ck+1(λ)) + c2 k−1 

(M 1k 

+ 

c11 

12...r−1r 

12...r−1k 

(Ck+1)) 

2 

M 12...r−1 

12...r−1 

(Ck+1)M12...r 12...r (Ck+1(λ)) 

 

× M 12...k 

12...k Ck+1(λ) + (M12...k 

12...k 

(Ck+1)) 

2 

M 12...k−1 

12...k−1 (Ck+1) − ˆλk+1M 12...k 

12...k (Ck+1). 

Finally we have the following expression for I k+1 

k+1 (λ) with corresponding positive constants ar, 

2 r k − 1 (depending on k) and b1 ∈ R: 

I k+1 

k+1 (λ) = 

= 

 

 

k−1 

a1 + 

r=2 

k−1 

ar 

a1 + 

Gr(λ) 

r=2 

ar 

M12...r 12...r (Ck+1(λ)) 

 

 

Gk(λ) + b1. 

M 12...k 

12...k 

By (37) of Lemma A.7 we conclude that for λr 0, 2 r k, holds 

∂I k+1 

k+1 (λ) 

∂λp 

∂I k+1 

k+1 (λ) 

∂λk 

= 

 

∂Gk(λ) 

= a1 

∂λp 

k−1 

a1 + 

r=2 

k−1 

+ 

r=2 

ar 

 

ar ∂Gk(λ) 

0, 

Gr(λ) ∂λk 

∂ 

∂λp 

Ck+1(λ) + b1 

Gk(λ) 

0, 2 p k. (78) 

Gr(λ)


Remark C.2. In fact ∂I k+1 

k+1 (λ)/∂λp > 0, 2 p k, forλ = (λr) k+1 

r=1 ∈ Rk+1 , λr 0, 1 r 

k + 1, since by (38) we have ∂Gk(λ)/∂λp = A p p(C(λ ]p[ )) > 0. 

Let us suppose that I k k (0) = 0, i.e. 

0 = I k k (0) = M12...k−1 

12...k−1 

For k = 3, k = 4 and k = 5wehave 

c 2 1k−1 

c11 

+ (M12...k−3k−2 

12...k−3k−1 )2 

M 12...k−3 

12...k−3 M12...k−2 

12...k−2 

I 3 3 (0) = M12 12 

I 4 4 (0) = M123 123 

I 5 5 (0) = M1234 1234 

c 2 14 

c11 

c 2 13 

c 2 12 

c11 

c11 

+ (M12 

14 )2 

c11M 12 

12 

We prove that I k+1 

k+1 (0) = 0. Indeed, we get 

I k+1 

 

c2 1k 

k+1 (0) = M12...k 12...k 

c11 

+ (M12 

1k−1 )2 

c11M 12 

12 

+ (M123 

12k−1 )2 

M 12 

12 M123 

123 

+ M12...k−1 

12...k−1 

M 12...k−2 

− ck−1k−1 

12...k−2 

+ M12 

 

12 

− c22 = 0, 

c11 

+ (M12 

13 )2 

c11M 12 

12 

+ (M123 

124 )2 

+ (M12 

1k )2 

c11M 12 

12 

+ (M12...k−2k−1 

12...k−2k ) 2 

M 12...k−2 

12...k−2 M12...k−1 

12...k−1 

+ M123 

123 

M12 

− c33 , 

12 

M 12 

12 M123 

123 

+··· 

 

. 

+ M1234 

1234 

M123 − c44 

123 

+ (M123 

12k )2 

M 12 

12 M123 

123 

+··· 

+ M12...k 

12...k 

M 12...k−1 

− ckk 

12...k−1 

Since by Corollary A.5 we have 

 

 

 

A 

 

k−1 

k−1 (Ck) A k−1 

k (Ck) 

Ak k−1 (Ck) Ak k (Ck) 

 

 

 

= A∅ k−1k 

∅ (Ck)Ak−1k (Ck) or 

 

 

 

A 

 

k−1 

k−1 (Ck) A k−1k 

k−1k (Ck) 

 

 

 

= A k k−1 (Ck) 2 , 

A ∅ ∅ (Ck) A k k (Ck) 

we conclude that 

 

M 

 

 

12...k−1 

12...k−1 (Ck) M 12...k−2 

12...k−2 (Ck) 

M12...k 12...k (Ck) M 12...k−2k 

12...k−2k (Ck) 

 

 

 

= M 12...k−2k−1 

12...k−2k (Ck) 2 . 

Hence 

(M 12...k−2k−1 

12...k−2k (Ck)) 2 

M 12...k−2 12...k−1 

12...k−2 (Ck)M12...k−1 12...k (Ck) 

M 12...k−1 

12...k−1 

(Ck) + M12...k 

 

. 

 

. 

12...k−2k (Ck) 

M12...k−2k 

= 

(Ck) M 12...k−2 

, 

(Ck) 

12...k−2

and 


I k+1 

 

c2 1k 

k+1 (0) = M12...k 12...k 

c11 

+ (M12 

1k )2 

c11M 12 

12 

+ (M12...k−3k−2 

12...k−3k ) 2 

M 12...k−3 

12...k−3 M12...k−2 

12...k−2 

+ (M123 

12k )2 

M 12 

12 M123 

123 

+ M12...k−2k 

12...k−2k 

M 12...k−2 

12...k−2 

+··· 

− ckk 

If we change k with k − 1 in the last expression we obtain the right-hand part (up to a positive 

factor) of the expression for I k k (0). 

Finally we have proved (77) for I k+1 

k+1 (λ). Let us consider the function 

We have 

i k+1 

k+1 

by (78) and Remark C.2. So 

k+1 

(0) = Ik+1 (0) = 0 and 

i k+1 k+1 

k+1 (t) = Ik+1 (t ˆλ), t ∈ R. 

di k+1 

k+1 (t) 

dt 

= 

k 

p=2 

I k k >Ik k (ˆλ) = i k k (t) t=1 0. 

∂I k+1 

k+1 (λ) 

∂λp 

 

. 

ˆλp > 0 

We recall (see (32)) that for λ = (λ1,...,λm) ∈ C m and 1 k m we denote 

Using (27) 

we get 

λ [k] = (0,...,0,λk+1,...,λm), λ {k} = (λ1,...,λk, 0,...,0). 

Gm(λ) = A ∅ ∅ Cm(λ) = 

A k k Cm(λ) = 

 

∅⊆δ⊆{1,2,...,m} 

 

∅⊆δ⊆{1,2,...,k−1,k+1,...m} 

If we put Cm(λ [k] ) = Cm + m r=k+1 λrErr in (79) we get 

A k [k] 

k Cm λ 

= 

∅⊆δ⊆{k+1,k+2,...,m} 

Similarly, if we put Cm(λ) = Cm(λ {k} ) + m r=k+1 λrErr we get 

A k k Cm(λ) = 

 

∅⊆δ⊆{k+1,k+2,...,m} 

λδA δ δ (C), 

λδA k∪δ 

k∪δ (Cm). (79) 

λδA k∪δ 

k∪δ (Cm). (80) 

λδA k∪δ 

{k} 

k∪δ Cm λ . (81)


Using (76) we have 

fk ˆλkA k k (Ck) A k k Ck(ˆλ) −1 = ˆλkA kk+1...m 

kk+1...m (Cm) A kk+1...m 

kk+1...m Cm(ˆλ) −1 hence I k m = fkA k k (Cm(ˆλ)) − ˆλkA k k (Cm(λ [k] )) I k m (ˆλ), where the function I k m (ˆλ) is defined by 

kk+1...m 

Akk+1...m Cm(ˆλ) −1 kk+1...m 

A 

∅⊆δ⊆{k+1,k+2,...,m} 

 

Cm(ˆλ) − ˆλkA k 

k Cm ˆλ [k] 

 

 

I k m (ˆλ) := ˆλk 

kk+1...m (Cm)A k k 

 

= ˆλk 

kk+1...m 

Akk+1...m Cm(ˆλ) 

−1 A 

 

 

kk+1...m 

kk+1...m (Cm) Ak k (Cm(ˆλ [k] )) 

A kk+1...m 

kk+1...m (Cm(ˆλ)) Ak k (Cm(ˆλ)) 

 

 

(80), (81) 

= ˆλk 

kk+1...m 

Akk+1...m Cm(ˆλ) −1 

 

A 

× 

ˆλδ 

 

kk+1...m 

kk+1...m (Cm) Ak∪δ k∪δ (Cm) 

A kk+1...m 

kk+1...m (Cm(ˆλ)) Ak∪δ k∪δ (Cm(ˆλ {k} 

 

 

)) 

. 

Using (26) or (27) we conclude for λ = (0,λ2,...,λm) ∈ C m 

Finally we obtain 

I k m (ˆλ) = ˆλk 

A kk+1...m 

kk+1...m Cm(λ) = 

A k∪δ 

{k} 

k∪δ Cm λ = 

× 

kk+1...m 

Akk+1...m Cm(ˆλ) −1 

∅⊆γ ⊆{2,3,...,k−1} 

 

∅⊆γ ⊆{2,3,...,k−1} 

 

∅⊆γ ⊆{2,3,...,k−1} 

 

 

ˆλγ 

 

 

 

γ ∪{k,k+1,...m} 

λγ Aγ ∪{k,k+1,...m} (Cm), 

 

∅⊆δ⊆{k+1,k+2,...,m} 

A kk+1...m 

kk+1...m (Cm) A 

A k∪δ 

k∪δ (Cm) A 

γ ∪{k}∪δ 

λγ Aγ ∪{k}∪δ (Cm). 

ˆλδ 

γ ∪{k,k+1,...m} 

γ ∪{k,k+1,...m} (Cm) 

γ ∪{k}∪δ 

γ ∪{k}∪δ (Cm) 

 

 

 

0 

 

due to the Hadamard–Fisher’s inequality (Lemma A.6), for α ={k,k + 1,...,m} and β = γ ∪ 

{k}∪δ. This completes the proof of Lemma 16. ✷ 

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Finite range decompositions of positive-definite 

functions 

David Brydges a,∗ , Anna Talarczyk b 

a Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada 

b Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland 



Communicated by L. Gross 

Abstract 

We give sufficient conditions for a positive-definite function to admit decomposition into a sum of 

positive-definite functions which are compactly supported within disks of increasing diameters Ln .More 

generally we consider positive-definite bilinear forms f → v(f,f ) defined on C∞ 0 .Wesayvhas a finite 

range decomposition if v can be written as a sum v = Gn of positive-definite bilinear forms Gn such 

that Gn(f, g) = 0 when the supports of the test functions f,g are separated by a distance greater or equal 

to Ln . We prove that such decompositions exist when v is dual to a bilinear form ϕ → |Bϕ| 2 where B is 

a vector valued partial differential operator satisfying some regularity conditions. 


Keywords: Positive-definite; Generalized Gaussian field; Renormalisation group; Elliptic operator; Green’s function 


Let Λ be an open set in Rd , which may be all of Rd , and let D(Λ) = C∞ 0 (Λ). Letf,g ↦→ 

v(f,g) be a bilinear form defined for f,g ∈ D(Λ). The form v is said to be positive-definite if 

v(f,f ) > 0 for every non-vanishing f ∈ D(Λ). 


E-mail addresses: db5d@math.ubc.ca (D. Brydges), annatal@mimuw.edu.pl (A. Talarczyk). 


doi:10.1016/j.jfa.2006.03.008

D. Brydges, A. Talarczyk / Journal of Functional Analysis 236 (2006) 682–711 683 

Given positive-definite continuous bilinear forms, v and Gj 

,j ∈ N, onD(Λ), we write v = 

j1 Gj when 

v(f,f ) = 

Gj (f, f ) 

j1 

holds for all f ∈ D(Λ). 

The forms v, Gj are said to be translation invariant if Λ = Rd and v(Ttf,Ttg) = v(f,g) for 

all t ∈ Rd , where Ttf(x)= f(x+ t). We often encounter the case where the bilinear form arises 

from a function ˜v(x − y) such that 

 

v(f,g) = f(x)˜v(x − y)g(y)dx dy. 

˜v is said to be a positive-definite function when the form v is positive-definite. 

Definition 1.1. Let v be a translation invariant bilinear form. We say that v admits a translation 

invariant finite range decomposition if, for some L>1, there exist positive-definite forms Gj 

such that 

(1) v = 

j1 Gj ; 

(2) Gj (f, g) = 0ifdist(supp f,supp g) Lj ; 

(3) for j ∈ N, Gj is translation invariant. 

This paper is concerned with the question of when a bilinear form has a finite range decomposition. 

Our main result on the existence of such decompositions is given in Theorem 2.6. It 

concerns bilinear forms associated to the Green’s functions of a large class of elliptic partial 

differential operators. It also gives a decomposition if v is not translation invariant and then con- 

dition (3) is replaced by a kind of uniformity of convergence of 

j1 Gj . Theorem 2.8 and 

Proposition 2.9 give a more explicit form of this decomposition. In Section 4 we give concrete 

examples based on the construction used to prove existence in Section 3. 

Our interest is motivated by an equivalent question concerning Gaussian random fields. It 

is well known (e.g., see [9]) that for any continuous bilinear form v there exists a generalized 

Gaussian random field, i.e. a distribution valued random variable φ such that for any 

test functions ϕ1,...,ϕn ∈ D(Λ) the vector (〈φ,ϕ1〉,...,〈φ,ϕn〉) is centered Gaussian and 

Cov(〈φ,ϕ〉, 〈φ,ψ〉) = v(ϕ,ψ). The existence of a finite range decomposition of v is equiva- 

lent to the existence of a decomposition φ = 

j1 ζj , where ζj are independent generalized 

Gaussian random fields with covariance functionals Gj respectively and such that 〈ζj ,ϕ〉 and 

〈ζj ,ψ〉 are independent if dist(supp ϕ,supp ψ) L j . 

Many models in statistical mechanics have the form of an expectation EZ of a nonlinear 

functional Z of a Gaussian random field φ, where φ has long range power law correlations and 

the functional Z depends on the field φ in a large region Λ ⊂ R d . The large size of Λ and the 

long range correlations make it difficult to get accurate estimates on EZ. The class of methods 

known as the Renormalisation Group (RG) was originated by K.G. Wilson [12,13] in order to 

address this problem. 

A very convenient version of Wilson’s idea was introduced in rigorous mathematical arguments 

by Gallavotti et al. in [1,2]. These authors write the field φ = 

j1 ζj as a sum of

684 D. Brydges, A. Talarczyk / Journal of Functional Analysis 236 (2006) 682–711 

independent increments ζj so that Z becomes a function Z = Z1(ζ1 + ζ2 +···) of all the increments 

and then they define the conditional expectation Ej to be the integration over increment ζj . 

E.g., let Zj+1 = Ej Zj so that EZ = lim Zj . The paper [1] is a good introduction to the RG. 

In essentially all papers based on [2] and related decompositions introduced by other authors, 

e.g., [6,7], the decay of the ζj covariance is exponential on scale L j but not finite range as in 

property (2) above. The machinery known as “cluster expansions” was developed to control this 

lack of exact independence and these expansions have become the workhorse of rigorous work 

on the RG. Thus finite range decompositions are certainly not essential for progress in the RG, 

but by removing the need for cluster expansions they replace a major technical prerequisite of 

this subject. Wavelet decompositions are also used in the RG [5]. These decompositions can 

have the finite range property but not independence of ζi and ζj for j = i because they are not a 

decomposition of the covariance into a sum of positive definite forms. 

These considerations have motivated us to write this paper to prove that finite range decompositions 

exist for a wide class of Gaussian fields. In the case that the covariance of the Gaussian 

field φ is homogeneous, e.g., |x − y| −α , it is easy to establish existence of these decompositions 

and we have made this point and used them, for example, in [3]. In [4] we proved that the resolvent 

of the Laplacian (aI − ) −1 with a 0 admits finite range decompositions and also 

showed that these decompositions exist when is the finite difference Laplacian on the simple 

cubic lattice Z d . 

Finite range decompositions for radial functions have appeared for different reasons in the 

context of the stability of matter. In [8] are given necessary and sufficient conditions on the 

derivatives of a function f that defines a radially symmetric kernel f(x − y) so that the bilinear 

form associated to f(x − y) has an expansion with non-negative coefficients into tent 

functions. 

We defer precise definitions and give a little outline of our results. Two bilinear forms v and 

E are said to be dual if the Hilbert space completions of C ∞ 0 (Rd ) in the two bilinear forms 

are dual relative to the L 2 (R d ) inner product. Our main condition for v to admit a translation 

invariant finite range decomposition is that the dual bilinear form E is associated with a constant 

coefficient partial differential operator B by 

 

E (ϕ, ϕ) = 

R d 

|Bϕ| 2 dx, 

where B can be vector valued. B need not be first order. As an example consider 

so that 

 

E (ϕ, ϕ) = 

B = (∂1,...,∂d,λI) 

R d 

|∂ϕ| 2 + λ 2 |ϕ| 2 dx. (1.1) 

By integration by parts this form is associated to the elliptic partial differential operator B ′ B = 

− + λ 2 . The duality of v,E means that the distribution kernel of v is a Green’s function 

(B ′ B) −1 for the partial differential operator B ′ B.


Our condition does not characterise the forms that have translation invariant finite range decompositions 

because one can deduce from it that the kernel of (B ′ B) −α with α ∈ (0, 1] also has 

a finite range decomposition. We know of no examples of positive-definite forms that are not in 

this wider class. 

The construction in our proof achieves much more because it also creates decompositions 

satisfying items (1), (2) when B has non-constant coefficients and Λ need not be all of R d . 

In this case v is not translation invariant, hence (3) cannot be satisfied and is replaced by the 

following. If the partial differential operator B has constant coefficients, but the domain Λ is not 

all of R d then the decomposition satisfies: 

• Translation invariance away from ∂Λ.Forj ∈ N small enough such that 2L j < diam Λ, 

Gj (Ttf,Ttf) is independent of t and Λ for f,t such that the support of Ttf is in Λ but 

separated from the boundary of Λ by a distance greater than L j . 

In a few examples like the ones discussed in Section 4 where we can make explicit computations 

of norms, we find that the terms in our decompositions decay with a scaling that correctly 

reflects the dimensional analysis of the operator B ′ B. If the coefficients of B are not constant, 

we do not know very much about the rate of convergence of the decomposition. We only have 

the following estimate which says that the decomposition converges uniformly with respect to 

translation of the argument f : 

• Uniformity. There exists a constant c such for all L>cthere exist finite range decompositions 

such that for all f = B ′ Bϕ in B ′ BD(Λ), 

0 v(f,f ) − 

jn 

(n−p)∨0 

c 

Gj (f, f ) 

v(f,f ), (1.2) 

L 

where p is the smallest integer such that diam supp ϕ L p . The class B ′ BD(Λ) is dense in 

the Hilbert space with inner product v. 

We examine these decompositions for the simple case of the Laplacian in Section 4 in order 

to understand these decompositions more concretely and in particular to examine their smoothness. 

In Section 5 we continue these calculations for the Laplacian to construct a finite range 

decomposition with C ∞ smoothness. 

2. Notation and main result on existence 

The proofs of the results in this section are found in Section 3. 

Let B = (B1,...,Bn) be an n-vector of partial differential operators, 

Bi = 

ci,α(x)∂ α . 

We call 

E (ϕ, ψ) = 

i=1 

a Dirichlet form. We impose the following assumptions. 

α 

n 

 

BiϕBiψdx= (Bϕ, Bψ) L2 (2.1)


Assumptions. 

(1) For each i, α, ci,α ∈ C∞ (Λ). 

(2) For each ψ ∈ D(Λ) there exists c such that for all ϕ ∈ D(Λ) 

 

 

ψϕdx 

c E (ϕ, ϕ) 1/2 . (2.2) 

(3) The continuity Hypothesis 2.2 explained below. 

By the first assumption Bϕ is defined for all ϕ ∈ D(Λ) and 

B ′ ψ = 

i,α 

(−∂) α (ci,αψi) 

is defined for ψ any vector-valued smooth function. Note that this implies that for any ϕ ∈ D(Λ) 

B ′ Bϕ ∈ D(Λ). By integration by parts (Bϕ, ψ) L 2 = (ϕ, B ′ ψ) L 2. 

By the second assumption, with ϕ = ψ, E (ψ, ψ) = 0 implies ψ = 0 and so E is an inner 

product defined on D(Λ). LetH+(Λ) be the Hilbert space completion of D(Λ) with inner 

product E . The corresponding norm will be denoted by ·+. For any open subset U ⊂ Λ, let 

H+(U) be the closed subspace of H+(Λ) obtained by taking the closure of D(U). 

Definition 2.1. We say that H+(U) is continuous at U if 

H+(V ): V ⊃ U = H+(U). 

The last of the three hypotheses mentioned above is 

Hypothesis 2.2. Λ is convex and H+ is continuous at U for all sets U which are bounded, convex 

and open. 

This is an implicit condition on the coefficients of the partial differential operator B. Asufficient 

condition for Hypothesis 2.2 to hold is provided by the following lemma. 

Lemma 2.3. Let U be a bounded convex open set. H+ is continuous at U if, there exists x∗ ∈ U, 

δ>0, C>0 and a convex open Ũ ⊃ U, Ũ ⊂ Λ such that |ci,α(x∗ + λ(x − x∗))| C|ci,α(x)| 

for all x ∈ Ũ,alli, α and all λ ∈[1 − δ,1]. 

Remark 2.4. In particular, Hypothesis 2.2 holds if the coefficients ci,α are bounded away from 

zero in every bounded subset of Λ. 

Let H−(Λ) be the abstract dual space of bounded linear functionals on H+(Λ). H−(Λ) is 

contained in the space of distributions 

H−(Λ) ⊂ D ′ (Λ)


because the topology on D(Λ) is stronger than the topology on H+(Λ) (a subsequence ϕn is 

convergent in the topology of D(Λ) if there exists a compact set K ⊂ Λ with supp ϕn ⊂ K and 

all derivatives ∂ α ϕn converge uniformly). Therefore 

where 

H−(Λ) = f ∈ D ′ (Λ): f −,Λ < ∞ , (2.3) 

f −,Λ = sup 〈f,ϕ〉: ϕ ∈ D(Λ), E (ϕ, ϕ) 1 . (2.4) 

Since the dual of a Hilbert space is an isomorphic Hilbert space, the norm arises from an inner 

product. 

Definition 2.5. Define G(f, g) = (f, g)−,Λ to be the inner product on H−(Λ). In particular, 

G(f, f ) =f 2 −,Λ . (2.5) 

By Assumption (2), a function ψ ∈ D(Λ) determines a linear functional fψ ∈ H−(Λ) by 

 

〈fψ,ϕ〉= ψϕdx 

and in this sense D(Λ) ⊂ H−(Λ) so that G is a positive-definite bilinear form on D(Λ). 

The main result of this paper is the following. 

Theorem 2.6. Under Assumptions (1)–(3) given above, G admits a decomposition satisfying 

(1), (2) in Definition 1.1 and the uniformity estimate (1.2). IfΛ = R d and if B has constant 

coefficients, then this decomposition for G is a translation invariant finite range decomposition. 

If the partial differential operator B has constant coefficients, but the domain Λ is not all of R d , 

then the decomposition is translation invariant away from ∂Λ as defined above. 

To understand why this is a decomposition of the Green’s function for the differential operator 

B ′ B we use a standard argument called the Friedrich’s extension [10, p. 278], [11, p. 177]. We 

will show that G is the form for a Green’s function for the partial differential operator B ′ B with 

zero boundary conditions on ∂Λ. 

By the definition of the + norm, for all ϕ ∈ D(Λ), ϕ2 + =Bϕ2 

L2. Therefore the closure 

¯B : H+(Λ) → L 2 Λ,R n 

of B is an isometry. Let ¯B ′ : L2(Λ, Rn ) → H−(Λ) be the dual operator and define L = ¯B ′ ¯B. 

This is a map from H+(Λ) to H−(Λ) and it satisfies 

Λ 

〈 ¯B ′ ¯Bϕ,ψ〉=(ϕ, ψ)+ 

for all ϕ,ψ ∈ H+(Λ). Therefore it is the Riesz isomorphism that identifies the Hilbert space 

H+(Λ) with the dual H−(Λ) and so G is related to the inverse of L by 

G(f, g) = (f, g)−,Λ = L −1 f,g . 

On the domain ϕ ∈ D(Λ) we can omit the closures so that L ϕ is our differential operator B ′ Bϕ.


Now we give some results explaining in more detail the construction of the decomposition of 

Theorem 2.6. 

For U any convex bounded open subset of Λ, letpU be the orthogonal projection in H+(Λ) 

onto the subspace H+(U). In particular pU = 0forU =∅.Weset 

Ux = (x + U)∩ Λ. 

Lemma 2.7. For each ϕ ∈ H+(Λ) there exists a unique vector Tϕin H+(Λ) such that 

(T ϕ, ψ)+ = 1 

 

|U| 

The linear operator T : H+(Λ) ↦→ H+(Λ) is a contraction. 

dx(pUx ϕ,ψ)+ for all ψ ∈ H+(Λ). (2.6) 

The main ingredient of the decomposition is the following theorem allowing to “cut out” from 

G a bilinear form which is positive-definite and of finite range. 

Theorem 2.8. Let U be a convex, bounded open subset of Λ.Forf ∈ H−(Λ), define 

Then the bilinear form G1 such that 

is: 

(1) positive-definite; 

(2) finite range with range 2diamU. 

A ′ U f = f − T ′ f. (2.7) 

G(f, f ) = G1(f, f ) + G A ′ U f,A ′ 

U f 

We construct the finite range decomposition by an iterated application of Theorem 2.8. Let 

U0 be a convex open bounded set containing the origin and for j = 1, 2,...let Uj be a sequence 

of domains obtained by scaling U, 

For each domain construct T ′ 

j 

Uj = x: L −j x ∈ U . 

(2.8) 

, A ′ 

j , ˜Gj by replacing U by Uj in the construction of G1 given 

above. Set f1 = f and for j 2, set fj = A ′ 

j−1 fj−1. Define bilinear forms Gj for j 1by 

Gj (f, f ) = ˜Gj (fj ,fj ). 


(1) G(f, f ) = n j=1 Gj (f, f ) +A ′ 

n ...A ′ 

1f 2 −,Λ . 

(2) Given L>1, let the diameter of U0 be chosen less than 1 

2 (1 − L−1 ). Then for all j 1 the 

range of Gj is less than Lj .


(3) Let U0 be as in part (2). Then there exists a constant c = c(U0)>1 such that for all n ∈ N, 

for all L>1, for all f ∈ B ′ BD(Λ), 

 

A ′ 

n 

′ 

...A 1f (n−p)∨0 

 

c 

 

f −,Λ, 

−,Λ L 

where f = B ′ Bϕ and p 0 is smallest integer such that the diameter of the support of ϕ is 

less than L p . 

(4) For all f ∈ H−(Λ) and therefore, in particular, f ∈ D(Λ), G(f, f ) = 

j1 Gj (f, f ). 

Let us discuss a bit more our operator AU , which is the key ingredient of our decomposition. 

Remark 2.10. Let U be a bounded open subset of Λ. The linear operator defined on H+(Λ) by 

PU = I − pU 

is called the Poisson operator of the domain U for the following reason. Consider ψ = PU ϕ, 

where ϕ ∈ H+(Λ). Then (a) ψ satisfies B ′ Bψ(x) = 0forx∈U, where B ′ B is the partial differential 

operator applied in the distribution sense, and (b) ψ(x) − ϕ(x) ∈ H+(U). Part (b) is 

obvious by the definition of PU . It implements the boundary condition on ψ that ψ = ϕ outside 

U. To check (a), let φ ∈ D(U) be a test function and recall that B ′ B on the domain D(U) 

is the Riesz isomorphism. Therefore 

 

ψB ′ Bφdx = (PU ϕ,φ)+ = (ϕ, PU φ)+ = 0. 

Λ 

For example, when B ′ B =− and U has a smooth boundary, conditions (a), (b) constitute 

solving the Dirichlet problem inside U with boundary conditions given by ϕ restricted to ∂U. 

Poisson kernels also provide another useful representation for the operator AU of Theorem 

2.8. 

Lemma 2.11. Assume that U is a bounded open set. Then 

AU ϕ = 1 

 

|U| 

R d 

(2.9) 

1x+U P(x+U)∩Λϕdx, (2.10) 

where P(x+U)∩Λ was defined in (2.9). IfP(x+U)∩Λϕ(y) is jointly measurable with respect to 

(x, y) then (2.10) can be understood as an integral defined pointwise, and we can write 

AU ϕ(y) = 1 

 

P(x+U)∩Λϕ(y)dx. (2.11) 

|U| 

y−U 

Remark 2.12. In the case L = B ′ B =−, or more generally, L = λ 2 I − , we can use this 

to give the following interpretation to the construction of G1. The inner product (f1,f2)−,Λ 

is the interaction energy of two distributions fi of electrostatic charge. Suppose that fi is an


approximate point mass distribution located at point Qi. The finite range of G1 can be understood 

in terms of replacing fi by another distribution ρi chosen so that (1) ρi is supported in a compact 

region Ki centred on Qi; (2) the potential outside Ki is unchanged. Then, for Qi sufficiently far 

apart that Ki do not overlap (f1,f2)−,Λ − (ρ1,ρ2)−,Λ = 0. 

The factor P ′ Ux fi constructs a charge distribution, with these properties, on the boundary of 

Ux ∋ Qi. Then AU spreads the charge distribution fi out by (1) spreading out the charge at Qi 

onto the boundary of Ux and (2) averaging over all Ux containing point A. This very specific 

choice of ρi achieves the difficult simultaneous goals of making G1 positive-definite and smaller 

than G(f, f ) and finite range. 

The advantage compared with our construction in [4] is that this particular average over Ux 

allows us to avoid reliance on translation invariance and to generalise so as to include Green’s 

functions of operators that are of higher than second order. 

3. Proofs for Section 2 

In order to prove Lemma 2.3 we need the following result. Define Dλϕ(x) = ϕ(x/λ) where 

ϕ ∈ D(V ) and ϕ is considered to be a function on R d by defining it to be zero outside V ⊂ Λ. 

Lemma 3.1. Let V be an open convex subset of Λ containing x∗ = 0. If there exist C1 > 0,δ>0 

such that |ci,α(λx)| C1|ci,α(x)|, for all i, α, λ ∈[1−δ,1] and x ∈ V , then Dλ uniquely extends 

to an operator Dλ : H+(V ) → H+(V ). The resulting family of operators is uniformly bounded 

in operator norm and is left strongly continuous in λ at λ = 1. 

Proof. Let ϕ ∈ D(V ). Then Dλϕ ∈ D(V ). Also 

E (Dλϕ,Dλϕ) = 

 

ci,α∂ α Dλϕ 2 dx = 

λ d−2|α| 

 

ci,α(λx)∂ α ϕ 2 dx 

i,α 

C(δ) 

 

ci,α(x)∂ α ϕ 2 dx = C(δ)E (ϕ, ϕ). 

i,α 

D(V ) is dense in H+(V ) so this proves that Dλ is uniformly bounded in operator norm and 

extends uniquely. By dominated convergence, 

i,α 

lim Dλϕ − ϕ+ = 0 

λ↑1 

for ϕ ∈ D(V ). By the uniform boundedness of Dλ strong continuity on a dense subset implies 

strong continuity. ✷ 

Proof of Lemma 2.3. We give the proof for case x∗ = 0. Let ϕ ∈ {H+(V ): V ⊃ U}. Since 

{H+(V ): V ⊃ U} ⊃H+(U), it suffices to prove that ϕ ∈ H+(U). LetUn be the open set 

of all points in Λ within distance 2 −n of U. Then there exists a sequence ϕn ∈ D(Un) that 

converges to ϕ in H+ norm. By Lemma 3.1, for 1 − δ λ


Proof of Theorem 2.6. This is a direct consequence of Proposition 2.9 so it suffices to prove the 

remaining results of Section 1. 

In order to show Lemma 2.7 and Theorem 2.8 we need some properties of the operators pU 

and T . Lemmas 3.3 and 3.5 show that operator T of (2.6) is well defined and is a contraction. 

Lemma 3.10 and Proposition 3.11 are essential for the proof that bilinear form G1 in (2.8) is 

positive definite and of finite range. 

Lemma 3.2. pU pV = 0 if U ∩ V =∅. 

Proof. First, note that if U,V are disjoint open sets, if ϕ ∈ H+(U) and ψ ∈ H+(V ), then 

(ψ, ϕ)+ = 0, because it is enough to prove this when ϕ ∈ D(U) and ψ ∈ D(V ) and then 

(ψ, ϕ)+ = (Bψ, Bϕ) L 2 = 0 because B is a partial differential operator (PDO). By this remark, 

for any ψ ∈ H+(Λ), (ψ, pU pV ϕ)+ = (pU ψ,pV ϕ)+ = 0. ✷ 

Lemma 3.3. If H+ is continuous at U and U is bounded, then pUx 

at x = 0. 

is strongly continuous in x 

Proof. We will give the proof in three steps (open and closed are defined as subsets of Λ with 

relative topology): 

(1) If Un, n = 1, 2,..., is a sequence of open sets such that for every compact set K ⊂ U, there 

exists n(K) such that Un ⊃ K for n n(K). Then pUnpU converges strongly to pU . 

(2) If H+ is continuous at U, ifUn,n= 1, 2,..., is a sequence of open sets such that for every 

open V ⊃ U, there exists n(V ) such that Un ⊂ V for n n(V ). Then pUn (I −pU ) converges 

strongly to 0. 

(3) If H+ is continuous at U and U is bounded, then pUx is strongly continuous in x at x = 0. 

(1) Let ϕ ∈ H+(Λ). We have to prove that pUnpU ϕ → pU ϕ in norm. Equivalently, we must 

prove pUnϕ → ϕ for any ϕ ∈ H+(U). It suffices to prove this for ϕ ∈ D(U), because pUn are 

uniformly bounded in operator norm. By the hypothesis on the sequence Un, Un contains the 

support of ϕ for all sufficiently large n and therefore pUnϕ = ϕ for all sufficiently large n. 

(2) It suffices to prove that pUnϕ+ → 0 for all ϕ ∈ H+(U) ⊥ . Every subsequence of pUnϕ has a weakly convergent subsequence, because the ball in H+(Λ) is weakly compact. Let φ be 

the limit of a subsequence. Then, by the hypothesis on Un, for any V ⊃ U, φ ∈ H+(V ), because 

H+(V ) is a subspace and therefore weakly closed. By the continuity hypothesis, Definition 2.1, 

φ ∈ H+(U). Therefore, along this subsequence 

pUn ϕ2 + = (pUn ϕ,ϕ)+ → (φ, ϕ)+ = 0. 

Since this is valid for every subsequence, pUn ϕ+ → 0. 

(3) Let ϕ ∈ H+(Λ) and let x → 0. We have 

pUx∩U ϕ = pUx∩U pU ϕ → pU ϕ by (1), 

pUx∪U ϕ − pU ϕ = pUx∪U ϕ − pUx∪U pU ϕ → 0 by(2). (3.1)


Notice that pUx∪U ϕ = pUx ϕ + ψx, where ψx ∈ H+(Ux) ⊥ and pUx ϕ = pUx∩U ϕ + ζx, where 

ζx ∈ H+(Ux). Therefore by orthogonality of ψx and ζx and then (3.1) we obtain 

ψx 2 + +ζx 2 + =ψx + ζx 2 + =pUx∪U ϕ − pUx∩U ϕ 2 + → 0. 

Hence ψx tends to zero and consequently, using (3.1) again, pUx ϕ → pU as x → 0. ✷ 

The support of Bϕ is contained in supp ϕ for ϕ ∈ D(Λ) because B is a PDO. Since any 

ϕ ∈ H+(U) can be approximated by a sequence ϕn with supp ϕn ⊂ U, 

¯BH+(U) ⊂ L 2 (U). (3.2) 

From now on we assume that U is a bounded open convex set and Λ is convex so that, by 

Hypothesis 2.2, H+ is continuous at Ux = (x + U) ∩ Λ for all x ∈ R d .Fixϕ ∈ H+(Λ) and 

define 

Fx = ¯BpUx ϕ. 

Since ¯B is an isometry, Lemma 3.3 implies that x ↦→ Fx is a norm-continuous map from R d 

to L 2 .AlsoFx is bounded in L 2 norm uniformly in x. Therefore (Fx,Fy) L 2 is continuous in 

(x, y), so that the following integrals are well defined or possibly positive infinite. 

Lemma 3.4. 

 

 

dx 

dy 

 

(Fx,Fy) L2 |U| 

where |U| denotes the Lebesgue measure of U. 

dx(Fx,Fx) L 2, 

Proof. Let ɛ>0. Choose disjoint cubes of side ɛ whose union equals R d and insert the partition 

of unity 1 = 1Δ, 

 

 

dx 

dy 

 

(Fx,Fy) L2 = 

 

dx 

 

 

dy 

 

Δ 

(Fx, 1ΔFy) L 2 

The sum over Δ was interchanged with the inner product using Fubini’s theorem and 

¯Fx(·)Fy(·) ∈ L1. 

By (3.2), we can insert 1Uy∩Δ=∅,Ux∩Δ=∅. Taking the sum over Δ outside the absolute values 

we bound by 

 

 

dx 

dy 

Insert the Cauchy–Schwartz inequality in the form 

Δ 

 

1Uy∩Δ=∅,Ux∩Δ=∅ 

(Fx, 1ΔFy) L2 (Fx, 1ΔFy) L 2 1 

2 (Fx, 1ΔFx) L 2 + 1 

2 (Fy, 1ΔFy) L 2. 

 

. 

 

 

 

.


The resulting integrand is jointly measurable and non-negative so integrals and sums can be put 

in any order. Therefore both terms give the same contribution and we bound by 

 

 

dx dy1Uy∩Δ=∅(Fx, 1ΔFx) L2. Δ 

Let |Uɛ|= dy1Uy∩Δ=∅. For the future, note that |Uɛ| tends as ɛ → 0tothevolume|U| of U. 

Then the preceding expression equals 

|Uɛ| 

 

 

dx(Fx, 1ΔFx) L2 =|Uɛ| dx(Fx,Fx) L2 and the lemma follows by taking ɛ → 0. ✷ 

Lemma 3.5. 

Δ 

 

dx (pUx ϕ,ψ)+ 

 

|U|ϕ+ψ+. 

Proof. It is sufficient to prove case ψ = ϕ because 

 

dx (pUx ϕ,ψ)+ 

 

 

= dx 

 

(pUx ϕ,pUx ψ)+ dxpUx ϕ+pUx ψ+ 

 

 

 

pUx ϕ2 + dx 

1/2 

pUy ψ2 + dy 

1/2 . 

Now we consider case ϕ = ψ for which there are no absolute values because (pUx ϕ,ϕ)+ 0. 

By the Cauchy–Schwartz inequality, 

 

 

 

 

 

λi(pUx ϕ,ϕ)+ = λipUx ϕ,ϕ λipUx ϕ, 

i i i 

i 

i 

+ i 

 

1/2 λj pUx ϕ ϕ+ 

j 

j 

+ 

 

 

1/2 = ¯λiλj (pUx ϕ,pUx ϕ)+ ϕ+. 

i j 

By considering Riemann sums, this implies 

 

dx(pUxϕ,ϕ)+ 

 

 

dx 

By Lemma 3.4, 

 

dx 

i,j 

dy(pUxϕ,pUy ϕ)+ 

 

= 

 

dx 

 

=|U| 

1/2 dy(pUxϕ,pUy ϕ)+ ϕ+. (3.3) 

 

dy(Fx,Fy) L2 |U| 

dx(pUxϕ,pUx ϕ)+ 

 

=|U| 

Putting this into (3.3), we obtain the lemma for the case ψ = ϕ. ✷ 

dx(Fx,Fx) L 2 

dx(pUx ϕ,ϕ)+.


Proof of Lemma 2.7. By the Riesz representation theorem and Lemmas 3.3 and 3.5, Tϕexists, 

the map ϕ ↦→ Tϕis linear and T 1. ✷ 

Remark 3.6. The integral defining T also exists in the stronger sense of Bochner integration [14, 

p. 132]. 

To prove Theorem 2.8 we are going to need some more properties of the operator T . 

Lemma 3.7. 

(T ϕ, T ϕ)+ (ϕ, T ϕ)+. 

Proof. Using (2.6) twice, first with ψ = Tϕwe have 

Hence by the definition of Fx, 

(T ϕ, T ϕ)+ = 1 

 

|U| 

(T ϕ, T ϕ)+ = 1 

 

|U| 

1 

 

|U| 

= 1 

 

|U| 

dx 1 

 

|U| 

dx 1 

 

|U| 

dy(pUx ϕ,pUy ϕ)+. (3.4) 

dy(Fx,Fy) L 2 

dx(Fx,Fx) L 2 by Lemma 3.4 

dx(pUx ϕ,ϕ)+. ✷ 

Let T ′ : H−(Λ) → H−(Λ) be the operator dual to T on the dual space H−(Λ) of bounded 

linear functionals on H+(Λ). This means 

〈T ′ f,ϕ〉=〈f,T ϕ〉. 

Lemma 3.8. The supports of T ′ f , A ′ f , Tf and A f are contained in the closure of 

 

x∈supp f Ux. 

Proof. For A ,T this follows easily from the definitions. Since A ′ = I − T ′ it suffices to consider 

T ′ .Letϕ be a test function with support outside the closure of 

x∈supp f Ux. Then 

〈T ′ f,ϕ〉=〈f,T ϕ〉= 1 

 

|U| 

because the integrand is zero if U x ∩ supp f =∅. ✷ 

dx〈f,pUx ϕ〉=0 

Likewise p ′ U : H−(Λ) → H−(Λ) is the dual of pU . Recalling the discussion following Theorem 

2.6, it is easy to verify that p ′ U = L pU L −1 and p ′ U is an orthogonal projection.


Lemma 3.9. For the operator p ′ U defined above we have: 

(1) p ′ U f = 0 if supp f ⊂ (U)c ; 

(2) p ′ U p′ V = 0 if U ∩ V =∅. 

Proof. (1) Let ϕ ∈ D(U). 〈p ′ U f,ϕ〉=〈f,pU ϕ〉=0 because supp pU ϕ ⊂ U. 

(2) Take the dual of Lemma 3.2. ✷ 

Lemma 3.10. The bilinear form: 

(1) (f, T ′ g)−,Λ has range diam U and is positive-definite; 

(2) (T ′ f,T ′ g)−,Λ has range 2diamU and is positive-definite. 

Proof. (1) By using the isomorphism L and (2.6) we have 

(T ′ f,g)−,Λ = 1 

 

|U| 

dx p ′ Ux f,g 

 

1 

= −,Λ |U| 

dx p ′ Ux f,p′ Ux g 

−,Λ . 

By Lemma 3.9 the integrand vanishes at x except when supp f and supp g both intersect the 

closure of Ux. Therefore, it is identically zero if the distance between supp f and supp g is larger 

than diam U. 

The above formula proves that (T ′ f,f )−,Λ 0. To prove that (T ′ f,g)−,Λ is positivedefinite, 

it suffices to prove that (T ϕ,ϕ)+ > 0 when ϕ = 0 because T ′ = L T L −1 . Since 

(Tϕ,ϕ)+ = 1 

 

|U| 

dx(pUxϕ,pUx ϕ)+ 

it is clear that the right-hand side is non-negative and vanishes iff pUx ϕ = 0 for all x. Suppose 

that pUx ϕ = 0 for all x.Letψ∈D(Ux) for some x. Then 

(ϕ, ψ)+ = (ϕ, pUx ψ)+ = (pUx ϕ,ψ)+ = 0. 

Therefore ϕ is orthogonal to the subspace generated by 

x D(Ux). By using a partition of unity 

any function in D(Λ) can be written as a sum of functions in 

x D(Ux) so ϕ is orthogonal to a 

dense subset and therefore is zero. 

(2) (T ′ f,T ′ g)−,Λ is clearly positive-definite. From the proof of Lemma 3.4, 

(T ′ f,T ′ g)−,Λ = 1 

 

|U| 

dy 1 

 

|U| 

dx p ′ Uy f,p′ Ux g 

−,Λ . 

The rest of the argument is similar to (1): the closures of Ux,Uy must intersect, otherwise the 

integrand vanishes by Lemma 3.9. Also by Lemma 3.9, Ux must intersect supp g and Uy must 

intersect supp f . ✷ 


(1) (T ′ f,T ′ f)−,Λ (f, T ′ f)−,Λ; 

(2) T ′ f −,Λ f −,Λ.


Proof. These follow from the same properties already established for T in Lemma 3.7 and below 

the definition (2.6). ✷ 

Proof of Theorem 2.8. Since A ′ U = I − T ′ , 

G1(f, f ) = G(f, f ) − G A ′ U f,A ′ 

U f 

= (f, f )−,Λ − A ′ U f,A ′ 

U f −,Λ 

= 2(f, T ′ f)−,Λ − (T ′ f,T ′ f)−,Λ 

(f, T ′ f)−,Λ, 

where the last bound is from Proposition 3.11. Thus G1 is positive-definite. It is finite range by 

Lemma 3.10. ✷ 

Lemma 3.12. 

 

A ′ U f −,Λ f −,Λ. (3.5) 

Proof. By Proposition 3.11, 

′ 

A U f,A ′ 

U f −,Λ = (f, f )−,Λ − 2(f, T ′ f)−,Λ + (T ′ f,T ′ f)−,Λ 

(f, f )−,Λ − (f, T ′ f)−,Λ 

= f,A ′ 

U f −,Λ . 

Therefore A ′ U f 2 −,Λ A ′ U f −,Λf −,Λ. ✷ 

In the next lemma we assume that U is an open bounded convex set that contains the origin 

and, for R>1 we define the dilated set UR ={y: 1 R y ∈ U}. Since dilation and translation 

preserve convexity, H+ is continuous at x + UR. We define T with U replaced by UR. 

Lemma 3.13. Let A = I − T and let Dϕ be the diameter of the support of ϕ, then there exists a 

constant C such that for all ϕ ∈ H+(Λ), 

A ϕ+ C Dϕ 

R ϕ+. 

Proof. We assume that R>Dϕ since, otherwise, the result follows from Lemma 3.12. Let V = 

UR and Vx = V + x. Then 

A ϕ = ϕ −|V | −1 

 

dxpVx∩Λϕ. 

By (3.2), ¯BpVx∩Λ = 1Vx ¯BpVx∩Λ. Also, since dx1Vx (y) is independent of y and equals |V |, 

¯BA ϕ =|V | −1 

 

dx1Vx ¯B(I − pVx∩Λ)ϕ.

Furthermore, 


1Vx ¯B(I 

 

0 if supp ϕ ⊂ Vx, 

− pVx )ϕ = 

0 if supp ϕ ⊂ Rd \ V x 

where we used (3.2) in the second case, so that 

A ϕ+ =¯BA ϕL2 |V | −1 

 

dx 1Vx ¯B(I − pVx )ϕ L2 =|V | −1 

 

dx 1Vx ¯B(I − pVx )ϕ L2 X(ϕ) 

|V | −1 X(ϕ) ϕ+, 

where X(ϕ) is the set of x such that 1Vx ¯B(I − pVx∩Λ)ϕ = 0. By the previous equation X(ϕ) is 

contained in the set of x such that Vx intersects both supp ϕ and the complement of supp ϕ. If 

x ∈ X(ϕ), then the smallest ball that covers Vx neither contains nor is disjoint from the smallest 

ball that covers supp ϕ. Therefore the distance between the centres of these balls lies in the 

interval [R − Dϕ,R+ Dϕ] and there are constants C1,C depending on U such that 

|V | −1 

X(ϕ) C1R −d (R + Dϕ) d − (R − Dϕ) d C Dϕ 

. ✷ 

R 

Proof of Proposition 2.9. (1) The first item follows immediately by induction using Theorem 

2.8 with U replaced by Uj ,j = 1, 2,...,n. 

(2) Given L, letU0 have diameter D 1 2 (1 − L−1 ). Recall from just before Proposition 2.9 

that f1 = f and, for j 2, fj = A ′ 

j−1 fj−1. By induction using Lemma 3.8, for j 2, 

 

j−1 

supp fj ⊂ y: dist(y, supp f) D 

We find, following the proof of Theorem 2.8, that 

Gj (f, f ) = 2 fj ,T ′ 

j fj 

 

−,Λ − T ′ 

j fj ,T ′ 

j fj 

 

−,Λ . 

k=1 

L k 

 

. (3.6) 

By Lemma 3.10, Gj (f, g) = 0 when the supports of fj and the analogously defined gj are 

separated by at least 2Lj D. Therefore, Gj has range 2D j k=1 Lk Lj . 

(3) Recall that L ϕ = B ′ Bϕ for ϕ ∈ D(Λ) and that L is the Riesz isometry. The operators 

Aj are self-adjoint operators on H+(Λ). It easily follows that A ′ 

j = LAjL −1 . Therefore 

 

A ′ ′ 

n ...A 1f =An ...A1ϕ+ =ϕn+1+, (3.7) 

−,Λ 

where we have defined ϕj inductively by setting ϕ1 = ϕ and, for j>1, ϕj = Aj−1ϕj−1. Since 

(3.6) says that the diameter of the support of fj is less than diam supp f + L j−1 and since 

Lemma 3.8 allows us to make exactly the same induction for ϕj , the diameter of the support of 

ϕj is bounded by diam supp ϕ + L j−1 which, for j − 1 p, is less than 2L j−1 because p 0is


defined so that diam supp ϕ L p . By Lemma 3.13, with U replaced by U0 and R = L j we have 

ϕj+1+ =Aj ϕj + CL −1 ϕj + for j − 1 p. By induction, 

ϕn+1+ C n−p L −(n−p) ϕp+ 

for n p. Combine this with (3.7) and Lemma 3.12 to obtain 

 

′ 

A n ...A ′ 

1f C − n−p L −(n−p) ϕ+. 

Since ϕ+ =L−1f −,wehaveprovedpart(3). 

(4) Since L = ¯B ′ ¯B is the isomorphism from H+(Λ) to H−(Λ) and since D(Λ) is dense in 

H+(Λ), thesetB ′ BD(Λ) is dense in H−(Λ). From (3) we know that the form R defined by 

R(f,f ) = G(f, f ) − 

Gj (f, f ) 

vanishes for f in the dense set B ′ BD(Λ). By(1),G(f, f ) R(f,f ) 0soR is bounded and 

therefore continuous on H−(Λ). Therefore R(f,f ) = 0 for all f . ✷ 

Proof of Lemma 2.11. Since p(x+U)∩Λϕ vanishes outside x + U, wehave 

AU ϕ = ϕ −|U| −1 

 

j1 

dx1x+U p(x+U)∩Λϕ =|U| −1 

 

dx1x+U (ϕ − p(x+U)∩Λϕ) 

which finishes the proof of (2.11). The second part of the lemma is obvious. ✷ 

4. Examples 

In this section we apply the general theory of the previous sections to the case B ′ B =− and 

obtain explicit formulas for the operator AU which is the key ingredient of our decomposition. 

In one dimension we also obtain a formula for AU when B ′ B is a diffusion operator. 

The calculations in this and the next section are motivated by a desire for insight into the 

smoothing properties of Ak. The bilinear form G defines a linear operator, f ↦→ G(f, ·) from 

H−(Λ) to H+(Λ). Using the same letters for forms and operators, the construction for Gj given 

above Proposition 2.9 is 

Gj = A1 ...Aj−1 

G − Aj GA ′ 

j 

′ 

A j−1 ...A ′ 

1 

for j 2. Thinking of A ′ 

j as A acting to the left we see that the integral operator kernel of G 

should be smoothed by the operators Ak, ifAk is smoothing. G1 = G − A1GA ′ 

j will not be 

smoother than G because it inherits a singularity on the diagonal from G. 

Throughout this section we set U = BL—the ball centered at 0 and of radius L. B is the 

standard unit ball. 

We first give the formula for the kernel of the operator A away from the boundary of the 

set Λ. 

(4.1)


Proposition 4.1. Assume that L>0, d 1, B =∇, and Λ is some bounded open set in R d , 

such that the set Λ−2L ={x ∈ Λ: dist(x, ∂Λ) > 2L} is nonempty. If y ∈ Λ−2L then ALϕ(y) = 

d 

R hL(y − x)ϕ(x)dx, where 

hL(x) = 

1 

|B|L 2 |x| d 

 

∂BL 

2x · z −|x| 2 1|x−z|LσL(dz), (4.2) 

where σL denotes the surface measure (normalized to 1) on a sphere of radius L. In the case 

d = 1, σL(dx) = 1 

2 (δ−L(dx) + δL(dx)) where δt(dx) denotes a unit point mass at x = t. 

Notice that since σL is a uniform measure on the sphere ∂BL the value of hL(x) depends only 

on |x|. 

Proof. From Theorem 2.8 and Lemma 2.11, for ϕ ∈ H+(Λ) and y ∈ Λ−2L we have 

By translation invariance, 

ALϕ(y) = 1 

|BL| 

ALϕ(y) = 1 

|BL| 

 

y+BL 

 

y+BL 

where ϕx(z) = ϕ(x + z). 

Recall the well-known formula for the Poisson kernel of the ball BL 

 

PBLϕ(y) = 

∂BL 

Px+BL ϕ(y)dx. (4.3) 

PBL ϕx(y − x)dx, (4.4) 

Ld−2 (L2 −|y| 2 ) 

|y − z| d 

ϕ(z)σL(dz). 

Note that this is valid also for d = 1. Applying this to (4.4), then changing the order of integration 

and substituting x = x ′ − z we obtain 

ALϕ(y) = 1 

 

|BL| 

∂BL 

= 1 

 

|BL| 

= 1 

|BL| 

 

= 

R d 

∂BL 

 

 

 

∂BL 

1x∈y+BL 

1x∈z+y+BL 

1z∈x−y+BL 

hL(y − x)ϕ(x)dx 

Ld−2 (L2 −|y − x| 2 ) 

|y − x − z| d 

ϕ(x + z) dx σL(dz) 

Ld−2 (L2 −|y − x + z| 2 ) 

|y − x| d 

ϕ(x)dx σL(dz) 

Ld−2 (L2 −|y − x + z| 2 ) 

|y − x| d 

σL(dz)ϕ(x) dx


with 

hL(x) = Ld−2 

 

|BL| 

= 

∂BL 

1 

|B|L 2 |x| d 

because |x − z| 2 =|x| 2 − 2x · z + L 2 . ✷ 

L 2 −|x − z| 2 

 

∂BL 

|x| d 

1|x−z|LσL(dz) 

2x · z −|x| 2 1|x−z|LσL(dz) 

The function hL of this proposition can be written in a more explicit form, namely: 

Proposition 4.2. In the notation of Proposition 4.1, ifd = 1 then 

hL(x) = 1 

 

1 − 

2L 

|x| 

 

1|x|2L. (4.5) 

2L 

If d = 2 then 

If d = 3 then 

hL(x) = 1 

π 2 L 2 

= 1 

π 2 L 2 

hL(x) = 

1 

|x|/2L 

2L 

√ 1 − r 2 

|x| 

r 2 

dr1|x|2L 

2 − 1 − arccos |x| 

 

1|x|2L. (4.6) 

2L 

3 

8πL|x| 2 

 

1 − |x| 

2 1|x|2L. (4.7) 

2L 

These expressions show that the kernel hL of AL is singular at x = 0, but the singularity is 

integrable. In Section 5 we will see that AL increases the smoothness (away from the boundary) 

by at least one derivative. 

Proof. Assume first that d = 1. Then the right-hand side of (4.2) equals 

1 

|B|L 2 |x| d 

 

∂BL 

2 

2xz − x 1|x−z|LσL(dz) = 1 

2L2 1 2 

2xz − x 

|x| 2 

1|x−z|L 

z=±L 

= 1 

4L2 2 

2L|x|−|x| 

|x| 

 

1|x−z|L = 

z=±L 

1 

 

1 − 

2L 

|x| 

 

2L 

1|x|2L. 

Case d 2. Referring to (4.2) let α be the angle between x and z so that x · z = L|x| cos α 

and the constraint |x − z| L is equivalent to 0 α arccos(|x|/2L). Then


hL(x) = 

1 

|B|L2 |x| d−1 

 

2L 

cos α −|x| 10αarccos(|x|/2L)σL(dz). 

For d 3 we can integrate over all points z ∈ ∂BL with α fixed. This set of points constitutes a 

(d − 2)-dimensional ball of radius L sin α. Keeping in mind that σL(dz) is normalized, we obtain 

hL(x) = 

2L 

|B|L 2 |x| d−1 

1 

cd−2 

 

arccos(|x|/2L) 

0 

 

cos α − |x| 

 

sin 

2L 

d−2 αdα (4.8) 

with cd−2 = π 

0 sind−2 αdα. This equation also holds for d = 2, because the normalised measure 

on the (d − 1) = 1-dimensional sphere is σL(dz) = 1 π dα with 0 α π and cd−2 = π. Putting 

d = 2 into (4.8) we obtain 

hL(x) = 

= 

2 

π 2 L|x| 

 

arccos(|x|/2L) 

0 

 

cos α − |x| 

 

dα 

2L 

2 

π 2 

sin 

L|x| 

arccos |x|/2L − arccos |x|/2L 

|x| 

2L 

= 1 

π 2L2 2 2L 

− 1 − arccos 

|x| 

|x|/2L 

. 

This finishes the proof for d = 2 since for r


Proposition 4.4. Let d = 1, B = d/dx, Λ = (a, b) and L>0, then the operator AL of Theorem 

2.8 is of the form ALϕ(y) = 

(a,b) aL(y, x)ϕ(x) dx, where 

⎧ 

0 for (y, x) /∈ (a, b) × (a, b), 

⎪⎨ y−a 

2L(x−a) for a


ALϕ(y) = 1 

 

2L 

a+L 

a−L 

+ 1 

 

2L 

y − a 

ϕ(x + L) 

x + L − a 1|x−y|L dx 

b+L 

b−L 

+ 1 

 

2L 

b−L 

a+L 

+ 1 

 

2L 

= 1 

2L 

 

b−L 

a+L 

a+2L 

a 

+ 1 

2L 

+ 1 

2L 

b − y 

ϕ(x − L) 

b − (x − L) 1|x−y|L dx 

ϕ(x − L) 

ϕ(x + L) 

x + L − y 

1|x−y|L dx 

2L 

y − (x − L) 

1|x−y|L dx 

2L 

y − a 

ϕ(x) 

x − a 1yx dx + 1 

2L 

 

b−2L 

a 

b 

a+2L 

ϕ(x) 

ϕ(x) 

which gives (4.10). 

If b − a


We now proceed to the question of regularity of ALϕ.Letϕ ∈ H+(Λ) then, in particular, ϕ is 

a continuous function on [a,b] which vanishes at a and b. Assume first that b − a>4L. By 

(4.14) we obtain that for y ∈ (a + 2L,b − 2L) 

ALϕ(y) = 

y 

y−2L 

From this we obtain for y ∈ (a + 2L,b − 2L) 

and 

2L − y + x 

ϕ(x) 

(2L) 2 

y+2L 

dx + 

d 

dy ALϕ(y) = 1 

(2L) 2 

 

− 

y 

y−2L 

y 

ϕ(x)dx + 

ϕ(x) 

 

y+2L 

y 

2L + y − x 

(2L) 2 

ϕ(x)dx 

 

dx. (4.15) 

(4.16) 

d2 dy2 ALϕ(y) = 1 

(2L) 2 

 

ϕ(y − 2L) + ϕ(y + 2L) − 2ϕ(y) . (4.17) 

If y ∈ (a, a + 2L) then, again by (4.14) we have 

and 

ALϕ(y) = 

y 

a 

+ 

2L − y + x 

ϕ(x) 

(2L) 2 

a+2L 

dx + 

 

y+2L 

a+2L 

ϕ(x) 

2L + y − x 

(2L) 2 

d 

dy ALϕ(y) = 1 

(2L) 2 

y 

− ϕ(x)dx + 

a 

 

a+2L 

y 

y 

y − a 

ϕ(x) 

2L(x − a) dx 

dx, (4.18) 

ϕ(x) 2L 

dx + 

x − a 

 

y+2L 

a+2L 

ϕ(x)dx 

 

(4.19) 

d2 dy2 ALϕ(y) = 1 

(2L) 2 

 

ϕ(y + 2L) − ϕ(y) − ϕ(y) 2L 

 

. (4.20) 

y − a 

Now it is easy to see, taking into account that ϕ(a) = 0, that if we let y → a + 2L all the 

corresponding limits in (4.15)–(4.20) coincide. Hence ALϕ is also twice continuously differentiable 

at y = a + 2L. The same reasoning shows that ALϕ is twice continuously differentiable 

in (b − 2L,b) and at b − 2L, which proves the statement of the proposition in case b − a>4L. 

The cases 2L b − a 4L and b − a 2L can be investigated in a similar way.


Now assume that f ∈ H−(Λ) ∩ Cb(Λ) and b − a>4L. We will show that A ′ Lf is twice 

continuously differentiable in (a + 2L,b − 2L) but does not have to be differentiable at a + 2L. 

(Similar argument can be applied to show that the same happens at b − 2L.) Note that 

A ′ Lf(x)= 

and by (4.10) we have that for x ∈ (a + 2L,b − 2L) 

A ′ L f(x)= 

x 

x−2L 

Similarly as in (4.16) we obtain 

R 

2L + y − x 

(2L) 2 

f(y)dy+ 

d 

dx A ′ 1 

Lf(x)= (2L) 2 

 

− 

x 

x−2L 

aL(x, y)f (y) dy 

 

x+2L 

x 

f(y)dy+ 

2L − y + x 

(2L) 2 

f(y)dy. 

 

x+2L 

x 

 

f(y)dy . (4.21) 

Clearly, the second derivative also exists in (a +2L,b−2L). On the other hand, if x ∈ (a, a +2L) 

then 

A ′ L f(x)= 

Hence for x ∈ (a, a + 2L) we have 

 

a 

x 

y − a 

2L(x − a) f(y)dy+ 

a 

 

x+2L 

x 

2L − y + x 

(2L) 2 

f(y)dy. 

d 

dx A ′ 1 

Lf(x)= (2L) 2 

x 

x+2L 

2L(y − a) 

− 

f(y)dy+ f(y)dy . (4.22) 

(x − a) 2 

Letting x → a + 2L in (4.21) and (4.22) we see that left and right derivatives of A ′ Lf at a + 2L 

are equal if and only if 

which is not true in general. ✷ 

 

a+2L 

a 

f(y)dy= 

 

a+2L 

a 

y − a 

2L f(y)dy, 

Proposition 4.4 can be extended to the case when the operator −B ′ B generates a onedimensional 

non-degenerate diffusion process. Operator AL can be written in terms of the scale 

function of the diffusion process. 

x


Proposition 4.6. Assume that d = 1, Λ = (a, b), B = c(x) d 

, where c is twice continuously 

dx 

differentiable and c2 (x) const > 0. Let s denote the scale function of the diffusion process 

generated by −B ′ B. Then the kernel of the operator AL is of the form: 

⎧ 

s(y)−s(a) 

2L(s(x)−s(a)) 

s(b)−s(y) 

for a


which coincides with (4.23). If b − a


Proposition 5.2. For each ϕ ∈ H+(R d ) the limit 

exists in H+(Λ) and 

Aϕ = lim 

n→∞ A L −n ...A L −2A L −1A1ϕ (5.4) 

A ϕ = h ∗ ϕ, (5.5) 

where h ∈ C∞ 0 (Rd ) and the support of h is contained in |x| DL 

L−1 . Moreover, 

G(f, g) = Γ0(f, g) + G(A ′ f,A ′ g), 

d 

f,g ∈ H− R , (5.6) 

where Γ0 is a positive-definite operator of finite range smaller than 2D + 2DL/(L − 1). 

By replacing A by A in the construction described above Proposition 2.9, we have 

Corollary 5.3. For D>0 there exists c(D) such that for each L c(D) the iteration of (5.6) 

generates C ∞ finite range decompositions of G. 

Proof of Proposition 5.2. Using the results of the previous section we have ALϕ = hL ∗ ϕ with 

hL given by (4.2). Hence 

A L −n ...A L −2A L −1A1ϕ = h L −n ∗···∗h L −1 ∗ h1 ∗ ϕ. 

Notice that by the definition of AL as an average of Poisson kernels we have that each hL 

integrates to 1. 

Let X0,X1,... be independent random variables with the same density h1. Then, by (4.9) 

L −k Xk has density h L −k . It is clear that ∞ k=0 L −k Xk converges almost surely to some random 

variable Y which has a density. Let us denote it by h. It is also clear that Y is bounded by 

DL/(L − 1), i.e. support of h is the ball centered at 0 and of radius DL/(L − 1). Notice that 

mk=0 L −k Xk has density ˜hm = h1 ∗···∗h L −m. Choose m such that 

m 

ˆhk(x) CL,n 

, (5.7) 

|x| d+2 

k=1 

we can do it by (4.9) and Lemma 5.1. (5.7) implies that ˜hm is continuously differentiable. 

From the definition of h it follows that 

h(x) = ˜hm ∗ L −(m+1)d h L −(m+1) · (x) (5.8) 

and since ˜hm is continuously differentiable, (5.8) implies that h is smooth. 

For n m we can write 

 

h1 ∗···∗hL−n(x) = E ˜hm x − 

n 

k=m+1 

L −k Xk 

 

.


By continuity of ˜hm we have 

lim 

n→∞ h1 

 

∗···∗hL−n(x) = E ˜hm 

−(m+1) 

x − L Y = h(x), 

where the last equality follows by (5.8). 

In a very similar way, possibly choosing larger m, we can show that also the derivatives of 

h1 ∗···∗hn converge to the derivatives of h and are bounded. From this we obtain that for any 

ϕ ∈ D(Rd ) we have (5.4) with Aϕ = h ∗ ϕ. Moreover, since for each n we have AL−n 1, 

(5.4) holds for any ϕ ∈ H+(Rd ) and A 1. 

To prove the other part of the theorem, let us denote 

G−k(f, g) = G(f, g) − G A ′ 

L−k f,A ′ 

L−kf d 

, f,g∈H−R , (5.9) 

and 

˜G−n(f, g) = G(f, g) − G A ′ 

1 ...A ′ ′ 

L−nf,A 1 ...A ′ L−ng d 

, f,g∈H−R . (5.10) 

Using (5.9) we can write 

˜G−n(f, g) = G−n(f, g) + 

n 

k=1 

′ 

G−(n−k) A 

L−(n−k+1) ...A ′ ′ 

L−nf,A L−(n−k+1) ...A ′ L−ng . (5.11) 

From Theorem 2.8 it follows that G−k is positive-definite for each k, hence so is ˜G−n.Fromthe 

same theorem we have that range G−k is 2DL −k and we also know that 

diam supp A ′ 

L −kϕ = diam supp ϕ + 2DL −k . 

Combining these two facts with (5.11) we obtain that ˜G−n has range smaller than 2D + 

2DL/(L − 1). By (5.4) 

G(f, g) − G(A ′ f,A ′ G) = lim ˜G−n(f, g). 

n→∞ 

Hence G(·,·) − G(A ′ ·,A ′ ·) is positive-definite and of range 2D + 2DL/(L − 1). ✷ 

Proof of Lemma 5.1. By (4.2) we have 

ˆh(y) = 

 

1 

 

e 

|x|2 

ix·y 

|B||x| d 

∂B 

σ(dz) 2x · z −|x| 2 1 2x·z|x| 2. 

Function h is symmetric, hence ˆh is real, so we can replace e ix·y by cos(x · y). We change the 

order of integration and substitute x = rw where r =|x| and w = x/|x| to obtain


ˆh(y) = |∂B| 

 

|B| 

∂B 

∂B 

 

σ(dz) 

∂B 

∂B 

σ(dw)1w·z>0 

 

2(w·z) 

0 

dr cos(rw · y) (2w · z) − r 

= |∂B| 

 

1 − cos(2(w · y)(w · z)) 

σ(dz) σ(dw)1w·z>0 

|B| 

(w · y) 2 

. 

By symmetry with respect to z we obtain 

ˆh(y) = |∂B| 

 

|B| 

 

σ(dz) 

1 − cos(2(w · y)(w · z)) 

σ(dw) 

2(w · y) 2 

, 

which is equal to (5.1). 

The estimate (5.2) is now obvious since by (5.1) 

∂B 

∂B 

ˆh(y) = sin2 y 

y 2 

( ˆh(y) can be also easily computed directly using (4.5)). 

The proof of (5.3) for d 2 is a little more involved. Clearly ˆh is bounded since h is integrable. 

Now assume that |y| > 1 and consider first the case d = 2. Let α be the angle between w and z 

and β the angle between y and w, then, using symmetries in (5.1), we have 

 

ˆh(y) = C 

π/2 

0 

 

dβ 

π/2 

Next we substitute u = cos α, v = cos β to obtain 

1 

 

ˆh(y) = C dv 

C1 

0 

 

1/2 

0 

0 

1 

0 

0 

dα sin2 (|y| cos β cos α) 

|y| 2 cos2 . 

β 

du sin2 (|y|u) 

|y| 2 v 2 

1 

√ 

1 − v2 √ 1 − u2 1 

1 

dv du 

1 + (|y|uv) 2 

u2 √ 

1 − u2 + C1 

|y| 2 

1 

1/2 

0 

1 

1 

dv du √ 

1 − v2 √ 1 − u2 0 

0 

C1 

∞ 

1 

1 


|y| 

1 + v2 u C2 C3 

√ + 

1 − u2 |y| 2 |y| , 

which proves (5.3) for d = 2. 

The proof for d 3 is easier. We start in the same way obtaining


ˆh(y) = Cd,1 

Cd,1 

Cd,2 

Cd,2 

|y| 

 

π/2 

0 

1 

0 

1 

0 

 

0 

 

dβ 

π/2 

0 

 

dv 

0 

0 

1 

dα sin2 (|y| cos β cos α) 

|y| 2 cos 2 β 

du sin2 (|y|uv) 

|y| 2 v 2 

1 

1 


u2 

1 + (|y|uv) 2 

∞ 

1 

dv 

1 + v2 which finishes the proof of (5.3). ✷ 


1 

0 

udu, 

sin d−2 α sin d−2 β 

The work of DB was supported in part by NSERC of Canada. AT is grateful to the University 

of British Columbia, Vancouver, Canada, for hospitality. 

References 

[1] G. Benfatto, N. Cassandro, G. Gallavotti, F. Nicolo, E. Olivieri, E. Presutti, E. Scacciatelli, Some probabilistic 

techniques in field theory, Comm. Math. Phys. 59 (1978) 143–166. 

[2] G. Benfatto, N. Cassandro, G. Gallavotti, F. Nicolo, E. Olivieri, E. Presutti, E. Scacciatelli, On the ultravioletstability 

in the Euclidean scalar field theories, Comm. Math. Phys. 71 (1980) 95–130. 

[3] D.C. Brydges, P.K. Mitter, B. Scoppola, Critical (Φ4 )3,ɛ, Comm. Math. Phys. 240 (1–2) (2003) 281–327. 

[4] D. Brydges, G. Guadagni, P.K. Mitter, Finite range decomposition of Gaussian processes, J. Statist. Phys. 115 (1–2) 

(2004) 415–449, http://arxiv.org/abs/math-ph/0303013. 

[5] P. Federbush, Quantum field theory in 90 minutes, Bull. Amer. Math. Soc. 17 (1987) 93–103. 

[6] K. Gawedzki, A. Kupiainen, A rigorous block spin approach to massless lattice theories, Comm. Math. Phys. 77 (1) 

(1980) 31–64. 

[7] K. Gawedzki, A. Kupiainen, Massless lattice ϕ4 4 theory: Rigorous control of a renormalizable asymptotically free 

model, Comm. Math. Phys. 99 (2) (1985) 197–252. 

[8] C. Hainzl, R. Seiringer, General decomposition of radial functions on Rn and applications to N-body quantum 

systems, Lett. Math. Phys. 61 (1) (2002) 75–84. 

[9] K. Itô, Foundations of Stochastic Differential Equations in Infinite-Dimensional Spaces, CBMS–NSF Regional 

Conf. Ser. in Appl. Math., vol. 47, SIAM, Philadelphia, PA, 1984. 

[10] M. Reed, B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, 

1972. 

[11] M. Reed, B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness, Academic 

Press, New York, 1975. 

[12] K.G. Wilson, J. Kogut, The renormalization group and the ɛ expansion, Phys. Rep. (Sect. C of Phys. Lett.) 12 (1974) 

75–200. 

[13] K.G. Wilson, The renormalization group and critical phenomena, Rev. Modern Phys. 55 (3) (1983) 583–600. 

[14] K. Yosida, Functional Analysis, sixth ed., Springer-Verlag, Berlin, 1980.



Weak semi-continuity of the duality product 

in Sobolev spaces 

Dorin Bucur 

Département de Mathématiques, UMR-CNRS 7122, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France 




Abstract 

Given a weakly convergent sequence of positive functions in W 1,p 

0 (Ω), we prove the equivalence between 

its convergence in the sense of obstacles and the lower semi-continuity of the term by term duality 

product associated to (the p-Laplacian of) weakly convergent sequences of p-superharmonic functions of 

W 1,p 

0 (Ω). This result implicitly gives new characterizations for both the convergence in the sense of obstacles 

of a weakly convergent sequence of positive functions and for the weak l.s.c. of the duality product. 


Keywords: Duality product; γ -Convergence; Obstacle convergence 


The limit of the scalar product of two weakly convergent sequences in a Hilbert space is, 

a priori, uncontrollable. In some particular situations, as for example in Sobolev spaces, when the 

sequences of functions are solutions (or supersolutions) of partial differential equations, extra information 

can be obtained on the scalar product of the limits by using qualitative properties of the 

solutions of the PDEs. In this paper, we are interested in duality products in the Sobolev spaces 

W −1,q × W 1,p 

0 involving p-superharmonic and positive functions. We characterize all sequences 

of positive functions, such that the duality product with the p-Laplacian of p-superharmonic 

functions is lower semi-continuous. 

E-mail address: bucur@math.univ-metz.fr. 


doi:10.1016/j.jfa.2006.03.014

D. Bucur / Journal of Functional Analysis 236 (2006) 712–725 713 

More precisely, let (un), (vn) ⊆ W 1,p 

0 (Ω) be two sequences of non-negative functions and 

assume they weakly converge to u and v, respectively. Assuming moreover that −pun 0in 

the sense of distributions on Ω, we wonder whether 

 

lim inf 

n→∞ 

Ω 

|∇un| p−2 

∇un∇vn dx 

Ω 

|∇u| p−2 ∇u∇vdx. (1) 

Under no further assumption, this assertion is in general false. For an extensive study of this 

question we refer the reader to [3] (see also [4] for further results), where the authors give sufficient 

conditions on the sequence (vn) in order that (1) holds true. Precisely, the main hypothesis 

is that the functions vn are also p-superharmonic −pvn 0. An example showing that the 

positivity condition vn 0 is not (in general) sufficient for the lower semi-continuity of (1) is 

also given for p = 2. The example relies on the emergence of the “strange term” appearing in 

the relaxation process of domains through γ -convergence (see [10]) and gives an intuitive hint 

on the fact that, in order that (1) is true, (vn) should vary such that their level sets do not produce 

relaxation measures via γ -convergence (see [5] for a detailed introduction to γ -convergence and 

Section 2 for a short review). 

The purpose of this paper is to give a characterization of all W 1,p 

0 (Ω)-weakly convergent 

sequences (vn) of non-negative functions for which (1) holds true for every W 1,p 

0 (Ω)-weakly 

convergent sequence (un) such that −pun 0. 

We prove that the necessary and sufficient condition that (vn) has to satisfy is to converge 

in the sense of obstacles (see Section 2 for the precise definition and [1,12] for details). This 

convergence is, in a certain sense, weaker than the strong convergence of W 1,p 

0 (Ω) and stronger 

than the weak convergence of W 1,p 

0 (Ω). Since (vn) is assumed by hypothesis weakly convergent 

in W 1,p 

0 (Ω), proving that it also converges in the sense of obstacles is equivalent to the possibility 

of finding a sequence θn strongly convergent to v in W 1,p 

0 (Ω), such that θn vn a.e. This is a 

consequence of the characterization of the obstacle convergence via the Mosco convergence of 

the convex sets 

Kvn = u ∈ W 1,p 

0 (Ω): u vn a.e. . 

In order to describe the obstacles, we make use on fine quasi-continuity properties of Sobolev 

functions. The proof of the main result of the paper relies on the characterization of the obstacle 

obst 

γ 

convergence vn −→ v in terms of the γ -convergence of the level sets {vn >t} −→ {v>t}, and 

on the knowledge of the relaxed measures associated to a γ -convergent sequence of quasi-open 

sets. 

Of course, the difficult part is the necessity. In [3], the hypothesis on the p-superharmonicity 

of vn insures the fact that min{vn,v} converges strongly to v in W 1,p 

0 (Ω)! So, taking θn = 

min{vn,v} we recover the obstacle convergence and fall into the sufficient part of the characterization 

result. 

All results in this paper hold for A-superharmonic functions, where −div A is a non-linear 

operator of p-Laplacian type. Precisely, assuming A : W 1,p 

0 (Ω) ↦→ W −1,q (Ω) is similar to the 

p-Laplacian (see the exact definition in Section 2) one can prove that if (vn) ⊆ W 1,p 

0 (Ω) is a 

weakly convergent sequence of non-negative functions, then vn converges in the sense of obsta- 

cles to the same limit v if and only if for every sequence of functions (un) ⊆ W 1,p 

0 (Ω), such that 

−div(a(x, ∇un)) 0 and un ⇀uweakly in W 1,p 

0 (Ω) we have

714 D. Bucur / Journal of Functional Analysis 236 (2006) 712–725 

 

lim inf 

n→∞ 

Ω 

 

a(x,∇un)∇vn dx 

Ω 

a(x,∇u)∇vdx. (2) 

For the simplicity of the exposition, our results are presented for the p-Laplace operator. We 

point out the fact that the convergence of (vn) into the sense of obstacles is independent on the 

choice of the operator −div(a(x, ·)). 

Section 2 contains a review of the main tools used in the paper, Section 3 contains the proof of 

the characterization result and the last section is devoted to some examples. A particular attention 

is given to uniformly oscillating obstacles. 

2. Obstacles, capacity and γ -convergence 

2.1. Capacity and relaxation measures 

is 

Let Ω ⊆ R N be a bounded open set and let 1 0 there exists a continuous 

(respectively lower semi-continuous) function fɛ : Ω → R such that capp ({f = fɛ},Ω)


(i) ∞|E(B) = 0 for every Borel set B ⊆ Ω with cap p (B ∩ E,Ω) = 0; 

(ii) ∞|E(B) =+∞for every Borel set B ⊆ Ω with cap p (B ∩ E,Ω) > 0. 

Definition 2.1. We say that a sequence (μn) of measures in M p 

0 (Ω) γp-converges to a measure 

μ ∈ M p 

1,p 

0 (Ω) if and only if Fμn : W0 (Ω) → R, 

Fμn (u) = 

Γ -converges in Lp (Ω) to Fμ, where 

 

Fμ(u) = 

 

Ω 

Ω 

|∇u| p 

dx + 

Ω 

|∇u| p 

dx + 

Ω 

|u| p dμn 

|u| p dμ. 

In order to simplify notations and since p is fixed, we drop the index p and instead of γp we 

note γ . 

We recall that Fn : W 1,p 

0 (Ω) → R Γ -converges to F in Lp (Ω) if for every u ∈ Lp (Ω) there 

exists a sequence un ∈ Lp (Ω) such that un → u in Lp (Ω) and 

Fμ(u) lim sup Fμn 

n→∞ 

(un), 

and for every convergent sequence un → u in L p (Ω) 

Fμ(u) lim inf Fμn (un). 

n→∞ 

In Definition 2.1, by the identification of a quasi-open set A with the measure ∞Ω\A, we 

implicitly have the definition of the γ -convergence of a sequence of quasi-open sets. In general, 

the γ -limit of a sequence of quasi-open sets is a measure of M p 

0 (Ω). In particular, this measure 

can be itself of the form ∞Ω\A. 

Note that the γ -convergence is metrizable by the following distance: 

 

dp(μ1,μ2) = |wμ1 − wμ2 | dx, 

where wμ is the variational solution of 

Ω 

−pwμ + μ|wμ| p−2 wμ = 1 (3) 

in W 1,p 

0 (Ω) ∩ Lp (Ω, μ) (see [5,15]). The precise sense of the equation is the following: wμ ∈ 

W 1,p 

0 (Ω) ∩ Lp (Ω, μ) and for every φ ∈ W 1,p 

0 (Ω) ∩ Lp (Ω, μ) 

 

Ω 

|∇wμ| p−2 

∇wμ∇φdx+ 

Ω 

|wμ| p−2 

wμφdμ= 

Ω 

φdx.


In view of the result of Hedberg [16], if A is an open subset of Ω, the solution of this equation 

associated to the measure ∞Ω\A is nothing else but the solution in the sense of distributions of 

−pw = 1 inA, w ∈ W 1,p 

0 (A). 

Throughout the paper, by wμ we denote the solution of (3) associated to the measure μ, and 

by wA the solution of the same equation associated to the measure ∞Ω\A. 

We refer to [14] for the following result. 

Proposition 2.2. The space M p 

0 (Ω), endowed with the distance dp, is a compact metric space. 

Moreover, the class of measures of the form ∞Ω\A, with A open (and smooth) subset of Ω, is 

dense in M p 

0 (Ω). 

Given a measure μ ∈ M p 

0 (Ω), we call the regular set of the measure μ the quasi-open set 

{wμ > 0}. We also notice that this set, which is denoted Aμ, coincides up to a set of zero capacity 

with the union of all finely open sets of finite μ measure. 

Lemma 2.3. Assume (An), (Bn) are two sequences of quasi-open sets which γ -converge to μA, 

μB, respectively. If capp (An ∩ Bn) = 0 for every n ∈ N, then cap(AμA ∩ AμB ) = 0. 

Proof. We notice that wAn · wBn = 0 q.e. Passing to the limit a.e. and using the quasi-continuity 

of wμA and wμB , we get wμA · wμB = 0 q.e., hence the conclusion. ✷ 

On M p 

0 (Ω), the following monotonicity is considered on the family of measures: 

μ1 μ2 if ∀A ⊆ Ω, A quasi-open, μ1(A) μ2(A). 

We notice (see [5,13]) that every monotone (increasing or decreasing) sequence of measures is 

γ -convergent. 

2.2. The obstacle problem 

Although the obstacle problem can be defined properly in the frame of measurable or quasilower 

semi-continuous functions, in the most part of the paper we restrict ourselves to obstacles 

which are elements of W 1,p 

0 (Ω). Roughly speaking, for q = p/(p − 1), given a function f ∈ 

W −1,q (Ω) and a measurable function v, the solution of the obstacle problem associated to v and 

f is the unique solution of the minimization problem 

where 

 

1 

min 

p 

Ω 

|∇φ| p 

−〈f,φ〉 

W −1,q 1,p : φ ∈ Kv , 

(Ω)×W0 (Ω) 

Kv = φ ∈ W 1,p 

0 (Ω): φ v a.e. Ω .


If the obstacle v is an element of W 1,p 

0 (Ω), the inequality φ v in the previous set can be 

equivalently taken in the sense a.e. or p-q.e. (for quasi-continuous representatives). In the sequel 

we concentrate our attention only on obstacles belonging to W 1,p 

0 (Ω). 

Definition 2.4. Let vn,v∈ W 1,p 

0 (Ω). We say that vn converges in the sense of obstacles to v if 

Γ -converges in L p (Ω) to 

W 1,p 

 

0 (Ω) ∋ u ↦→ 

Ω 

W 1,p 

 

0 (Ω) ∋ u ↦→ 

where ∞{v}(u) = 0ifu vp-q.e. and +∞ if not. We write vn 

Ω 

|∇u| p dx +∞{vn}(u) (4) 

|∇u| p dx +∞{v}(u), (5) 

obst 

−→ v. 

obst 

The main consequence of the convergence of obstacles vn −→ v is that for every f ∈ 

W −1,q (Ω) the sequence of solutions un of the obstacle problem associated to vn and f converges 

strongly in W 1,p 

0 (Ω) to the solution associated to v. 

The convergence in the sense of obstacles is equivalent to the convergence in the sense of 

Mosco of Kvn to Kv (see [1]), i.e. to the following two relations: 

1. ∀u ∈ Kv, ∃un ∈ Kvn such that un → u strongly in W 1,p 

0 (Ω). 

2. If unk ∈ Kvn 

1,p 

and unk ⇀uweakly in W 

k 0 (Ω), then u ∈ Kv. 

Notice that if vn converges weakly in W 1,p 

0 (Ω) to v, then the second Mosco condition is satisfied. 

We recall from [8, Corollary 4.9] and [12] the following characterization of the obstacle convergence. 

Theorem 2.5. Let vn,v ∈ W 1,p 

0 (Ω), vn 0. Then vn converges in the sense of obstacles to v if 

and only if there exists a dense set T ⊆ R such that 

Operators similar to the p-Laplacian 

γ 

{vn >t} −→ {v>t} ∀t∈T. Assume that a : Ω × R N → R N is a Carathéodory function which is homogeneous of degree 

p − 1 in the second variable 

a(x,tζ) =|t| p−2 ta(x, ζ ), t ∈ R, t= 0. (6) 

We notice that in order to have a precise description of the γ -limits of sequences of quasi-open 

sets, the homogeneity property is crucial (see [9,15]).


We suppose as usual the monotonicity assumptions on a(x,ζ) (see for instance [15]): there exist 

two constants c0,c1 with 0


 

= lim inf 

n→∞ 

Ω 

|∇un| p−2 

∇un∇θn dx = 

Ω 

|∇u| p−2 ∇u∇vdx. 

For the last equality, we notice, on the one hand, that |∇un| p−2∇un converges weakly to 

|∇u| p−2∇u in Lq (Ω, RN ) since −pun are positive and uniformly bounded Radon measures. 

This result is due to Boccardo and Murat [2] (see also [15, Theorem 2.10]) and is related to the 

pointwise convergence of the gradients. On the second hand, we have that θn converges strongly 

in W 1,p 

0 (Ω). 

Sufficiency: (ii) ⇒ (i). Assume (ii) holds. We shall prove that vn converges in the sense of 

obstacles to v. We rely both on the characterization of the obstacle convergence via the Mosco 

convergence, and since vn 0, on the characterization through the γ -convergence of the levelsets 

{vn >t}, Section 2, Theorem 2.5. 

We notice that from the weak convergence in W 1,p 

0 (Ω), vn ⇀v, the second Mosco condition 

M 

of the desired Mosco convergence Kvn −→ Kv is automatically satisfied. Moreover, the first 

Mosco condition has to be proved only for the function v. Indeed, if there exists θn vn such 

that θn → v in W 1,p 

0 (Ω)-strong, then for every ψ v, the first Mosco condition holds with the 

sequence min{θn,ψ} vn. 

Assume for contradiction that ∃δ>0 and a subsequence (vnk ) such that 

lim inf 

k→∞ min u − v 1,p δ>0. (13) 

u∈Kvn 

W0 (Ω) 

k 

In order to achieve the contradiction in assumption (13), we construct a subsequence of (vnk ), 

which converges in the sense of obstacles to v. 

Step 1. We construct a mapping 

[0, +∞) ↦→ μt ∈ M p 

0 (Ω), 

which is increasing in the sense of measures of M p 

0 (Ω) and such that for a subsequence of (vnk ) 

(denoted with the index r) and for a dense set T ⊆[0, +∞) we have 

∀t ∈ T {vr >t} −→ μt. 

The construction of the mapping t ↦→ μt is done by a diagonal procedure using the compactness 

and the metrizability of the γ -convergence in M p 

0 (Ω). We follow the approach used in [7] for 

constructing the relaxed space for obstacles. Let T = Q ∩ R + ={t1,t2,...,tk,...}. Forr = 1, 

..., we successively extract γ -convergent subsequences for the sequences of the quasi-open sets 

{vnk >tr}, and define μtr being the γ -limit. By a diagonal procedure, the metrizability of the 

γ -convergence gives the existence of a subsequence of (vn) and of a family of measures (μtr ) 

such that the level sets {vnk >tr} γ -converges to μtr . 

Using the density of the set T in [0, +∞) and relying both on the monotonicity of the 

measures and the γ -convergence of monotone sequences, we define the mapping t ↦→ μt by 

γ 

γ -continuity on the right. Finally, it can be concluded as in [8] that the convergence {vr >t} −→ 

μt holds on R \ N, where N is at most countable. Precisely, N is the set of discontinuity points 

of the monotonous real function [0, +∞) ∋ t → 

wμt Ω ∈ R. 

γ


By abuse of notation and for the simplicity of the exposition, we renote the constructed subsequence 

by (vn). 

We use the hypothesis (ii) and choose in relation (12) the sequence (un) defined in the following 

way: 

 

−pun = 0 in{vn >t}, 

(14) 

in Ω \{vn >t}. 

un = wΩ 

Following [17], we have −pun 0inΩ, so the sequence (un) is admissible in (ii). This 

sequence is bounded in W 1,p 

0 (Ω), hence for a subsequence, still denoted using the same index, 

we have that un ⇀uweakly in W 1,p 

0 (Ω). Following [2], we also have that ∇un →∇u a.e. in Ω, 

since −pun are uniformly bounded positive Radon measures. 

The information on the γ -convergence of the level sets {vn >t} gives: 

• u = wΩ on Ω \ Aμ. This is a consequence of the fact that u − wΩ ∈ L p (Ω, μ). 

• On Aμ, u satisfies the equation 

−pu + μ|u − wΩ| p−2 (u − wΩ) = 0 (15) 

in the variational sense of W 1,p 

0 (Ω) ∩ Lp (Ω, μ), i.e. for every φ ∈ W 1,p 

0 (Ω) ∩ Lp (Ω, μ) 

we have 

 

|∇u| p−2 

∇u∇φdx+ |u − wΩ| p−2 (u − wΩ)φ dμ = 0. (16) 

Ω 

Ω 

Indeed, for proving that u satisfies Eq. (15) on Aμ with the measure μ issued from the 

γ -convergence of the level sets {vn >t}, we can use a similar argument as in [9]. Denoting 

θn = un − wΩ and θ = u − wΩ, it is enough the prove that 

gn =: p(θn + wΩ) − pθn 

H 

−→ g =: p(θ + wΩ) − pθ, 

the convergence H being understood in the following sense: for every sequence 

φn ∈ W 1,p 

0 {vn >t} 

which converges weakly in W 1,p 

0 (Ω) to φ we have 

〈gn,φn〉 W −1,q (Ω)×W 1,p 

0 (Ω) →〈g,φ〉 W −1,q (Ω)×W 1,p 

0 (Ω) . 

Since we know that ∇un converges a.e. to ∇u, for every δ>0, there exists a set E such that 

|E|

lim 

 

n→∞ 

Ω 

lim sup 

n→∞ 

 

+ 


 

|∇un| p−2 ∇un −|∇θn| p−2 

 

∇θn ∇φn dx − 

E 

 

 

|∇un| p−2 ∇un −|∇θn| p−2 

∇θn ∇φn 

dx 

E 

 

|∇u| p−2 ∇u −|∇θ| p−2 ∇θ ∇φ dx. 

For every ε>0, there exists M such that 

Ω 

 

 

p−2 p−2 

|∇u| ∇u −|∇θ| ∇θ ∇φdx 

 

|∇ρ| M ⇒ ∇(ρ + wΩ) p−2 ∇(ρ + wΩ) −|∇ρ| p−2 ∇ρ ε|∇ρ| p−1 . 

Thus, for a given ε>0wehave 

 

 

|∇un| p−2 ∇un −|∇θn| p−2 

∇θn ∇φn 

dx 

E 

ε 

 

E∩{|∇θn|M} 

|∇θn| p−1 |∇φn| dx + 3M p−1 

C ε + 3M p−1 E ∩ |∇θn| t} , 

H 

that {vn >t} γ -converges to μ and gn −→ g. 

Applying inequality (12) to the sequence (un) constructed above, we get 

 

lim inf 

n→∞ 

|∇un| 

Ω 

p−2 

∇un∇vn dx |∇u| 

Ω 

p−2 ∇u∇vdx, 

or, decomposing the integrals by using vn − t = (vn − t) + − (vn − t) − , 

 

lim inf 

n→∞ 

Ω 

Ω 

|∇un| p−2 ∇un∇(vn − t) + 

dx − 

Ω 

|∇un| p−2 ∇un∇(vn − t) − dx 

 

|∇u| p−2 ∇u∇(v − t) + 

dx − |∇u| p−2 ∇u∇(v − t) − dx. 

Ω


We have that un = wΩ on {vn >t}, hence we get 

 

|∇un| p−2 ∇un∇(vn − t) − 

dx → |∇u| p−2 ∇u∇(v − t) − dx. 

Ω 

Consequently 

 

lim inf 

n→∞ 

Ω 

|∇un| p−2 ∇un∇(vn − t) + 

dx 

Ω 

Ω 

|∇u| p−2 ∇u∇(v − t) + dx. 

But, un is p-harmonic on {vn >t}, hence the integrals on the left-hand side are equal to zero! 

On the right-hand side, we use Eq. (16) satisfied by u on Aμ and get 

 

0 

Ω 

|u − wΩ| p−2 (wΩ − u)(v − t) + dμ. (17) 

Since all terms under the sum are positive on the right-hand side, we get 

 

|u − wΩ| p−2 (wΩ − u)(v − t) + dμ = 0. 

Ω 

By the comparison principle of p-superharmonic functions we know wΩ(x) > u(x) p-q.e. 

on Aμ. Consequently we get that μ({v>t}) = 0. 

The main idea is to prove that μ(Aμ) = 0 in which case μ =∞Ω\Aμ and thus Aμ ={v>t}. 

As a consequence we would get that the sequence of level sets {vn >t} γ -converges to {v >t} 

and Theorem 2.5 could be applied. It may be possible that {v >t} is a strict subset of Aμ (in 

the sense of capacity), so this argument is not enough to conclude that μ = 0onAμ, and needs 

further investigation. 

Step 2. From the weak-W 1,p 

0 (Ω) convergence vn ⇀v, we get 

(vn − t) + ⇀(v− t) + 

weakly in W 1,p 

0 (Ω), and from the γ -convergence of the sets {vn >t} to μ we get (v − t) + ∈ 

W 1,p 

0 (Ω) ∩ Lp (Ω, μ). 

The same holds also for (v − (t + ε)) + for ε>0. Let us denote 

At = γ − lim 

ε→0 {v>t+ ε}. 

This γ -limit exists from the monotonicity of the sets. Obviously we get At ⊆ Aμ. 

On the other hand, following Lemma 2.3, Aμ ∩{vt} we get that Aμ ={v>t} 

up to a set of zero capacity.


Step 3. We conclude the proof by noticing that the mapping t ↦→{v>t} is γ -continuous on R 

with the exception of an at most countable family of points, hence we get that μt =∞{vt} on R 

with the exception of an at most countable family of points, so the limit in the sense of obstacles 

of the sequence (vn) is the function v. ✷ 

Remark 3.2. In [3] the authors prove that if un,vn are weakly convergent sequences of nonnegative 

functions of W 1,p 

0 (Ω) such that −pun 0 and −pvn 0, then relation (1) holds 

true. The main technical argument is that min{vn,v} converges strongly in W 1,p 

0 (Ω) to v, which 

is obtained as a consequence of the p-superharmonicity of vn. We notice that the strong convergence 

min{vn,v}→v together with the weak convergence vn ⇀vin W 1,p 

0 (Ω) imply that 

obst 

−→ v, hence assertion (i) of Theorem 3.1 is satisfied. 

vn 

4. Further remarks 

The extension of the results of the paper to higher order operators is not an obvious matter. 

On the one hand, the positivity preserving property for higher order operators is depending on 

the geometric set where the operator is defined. For the bi-Laplace operator, positivity preserving 

holds, for example, on balls and the entire space but fails in general, even on smooth sets 

(see [11]). If the operators are positivity preserving, the sufficiency part of Theorem 3.1 still 

holds true. Nevertheless, dealing with the necessity part is more difficult, since the convergence 

of obstacles and the relaxation of sets through γ -convergence is not known. Moreover, the lack 

of reticularity of the Sobolev spaces of order greater than 1 may be a supplementary difficulty 

for the necessity part. 

Remark 4.1. There is a significant non-symmetry between the two terms in the duality product. 

The convergence in the sense of the obstacles of vn is related to the Mosco convergence of 

the sets Kvn and is independent on the choice of the operator itself which is associated to the 

terms un. This means that if Theorem 3.1 holds for a sequence (vn)n and the p-Laplace operator, 

then Theorem 3.1 holds for the same sequence (vn)n and an operator similar to the p-Laplacian. 

In the sufficient condition given in [3], the superharmonicity of the first sequence un serves 

for monotonicity of the duality product of −pun against vn and the metric projection of v on 

the cone {ϕ vn}, respectively, and on the other hand, the p-superharmonicity of the second 

sequence vn is a sufficient condition to obtain the obstacle convergence as a consequence of the 

weak convergence. 

Remark 4.2 (Uniformly oscillating obstacles). Assume that N p>N− 1 and vn ∈ W 1,p 

0 (Ω), 

vn 0 are continuous and have uniform oscillations, i.e. there exists a sequence of numbers (ln)n 

and a dense set {t1,t2,...}⊆R+ such that 

∀r, n ∈ N, ♯{vn tr} lr, (18) 

where ♯A denotes the number of the connected components of the set A. Ifvn⇀vweakly in 

W 1,p 

0 (Ω), then vn converges in the sense of obstacles to v. Indeed, the second Mosco condition 

is satisfied as a consequence of the weak convergence vn ⇀v. In order to prove the first Mosco 

condition, we follow the idea of the proof of Theorem 3.1, and assume for contradiction that 

relation (13) holds for some δ>0. From (18) and the shape compactness/stability result of [6],


there exists a subsequence of the sets {vn >tr} and a set Ωtr , such that {vnk >tr} γ -converges 

to Ωtr . Consequently, by a diagonal extraction procedure as in Theorem 3.1, we construct an 

obstacle h such that {h >tr} =Ωtr and the sequence vnk converges in the sense of obstacles 

to h.Fromtheγ-convergence of the level sets, we get (v − tr) + ∈ W 1,p 

(Ωtr 0 ), hence h v.This 

contradicts (13), from the Mosco convergence. 

It is needless to say that vn are not, in general, p-superharmonic. 

Remark 4.3. In [3], the authors give an example of a sequence of positive functions vn which 

are not superharmonic such that inequality (1) fails to be true for a suitable sequence un of superharmonic 

functions. This construction is done around the pioneering result of Cioranescu and 

Murat [10] on the “strange term” appearing in the relaxation process through the γ -convergence. 

The presence of the strange term is an argument of non-γ -convergence of obstacles! So, from 

this point of view it is not surprising that inequality (1) is violated, although the choice of un has 

to be done carefully. 

Remark 4.4. We give in the sequel an example of non-superharmonic functions vn for which 

the second assertion of Theorem 3.1 holds. Let η ∈ C1 (R2 , R) be a periodic function of period 

(l1,l2) and ϕ ∈ C∞ 0 (Ω). We consider the sequence of functions 

 

x 

vn = wΩ + ϕεη . 

ε 

It is clear that this sequence converges weakly but not strongly in H 1 0 (Ω) to v = wΩ, provided 

that ϕ or η are not the zero functions and ε ↓ 0. Inequality (1) is satisfied for all admissible 

sequences (un). This is a consequence of the obstacle convergence of vn towards v which can be 

easily proved. Nevertheless for small ε, the functions vn are not superharmonic! 

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Albeverio, Sergio, 634 

Barceló, Juan A., 1 

Bharali, Gautam, 351 

Bo˙zejko, Marek, 59 

Bryc, Włodzimierz, 59 

Brydges, David, 682 

Bucur, Dorin, 712 

Burq, Nicolas, 265 

Charles, L., 299 

Crabb, M.J., 630 

Cranston, M., 78 

Gamblin, Didier, 201 

Kissin, Edward, 609 

Kosyak, Alexandre, 634 

Koufany, Khalid, 546 

Kucerovsky, Dan, 395 

Li, Hong-Quan, 369 

Miller, Luc, 592 

Mountford, T.S., 78 

0022-1236/2006 Published by Elsevier Inc. 

doi:10.1016/S0022-1236(06)00209-6 

Journal of Functional Analysis 236 (2006) 726 

AUTHOR INDEX FOR VOLUME 236 

Olofsson, Anders, 517 

Planchon, Fabrice, 265 

Porretta, Alessio, 581 

Priola, Enrico, 244 

Ruiz, Alberto, 1 

Schultz, Hanne, 457 

Schulze, Michael, 120 

Serre, Denis, 409 

Talarczyk, Anna, 682 

Van Schaftingen, Jean, 490 

Vassout, Stéphane, 161 

Vega, Luis, 1 

Véron, Laurent, 581 

Wang,Feng-Yu,244 

Wi´snicki, Andrzej, 447 

Yakubovich, Dmitry V., 25 

Zhang, Genkai, 546 

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