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

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552 K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580 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
K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580 553 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
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Journal of Functional Analysis 236
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H.-Q. Li / Journal of Functional An
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x(s) = H.-Q. Li / Journal of Functi
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et et Donc, H.-Q. Li / Journal of F
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et Posons H.-Q. Li / Journal of Fun
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⎛ ⎜ ⎝ − H.-Q. Li / Journal
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Donc, H.-Q. Li / Journal of Functio
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5. Quelques problèmes ouverts H.-Q
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The assumption tells us that the fo
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3.4. The zeroes of ( √ z, η) D.
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4.1. Persistence of surface waves D
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When G = η ⊗ r, we obtain Becaus
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This polynomial satisfies D. Serre
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A. Wi´snicki / Journal of Function
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Therefore, A. Wi´snicki / Journal
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we now have (cf. (7.8)) that H. Sch
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2.2. Definition J. Van Schaftingen
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By duality, Kp,T is also defined by
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Using it, we obtain as in (3.2) tha
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5. Structure of the group ZS E. Kis
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The minimum is realized for S. Albe
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For m = 3wehave S. Albeverio, A. Ko
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hence S. Albeverio, A. Kosyak / Jou
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φpq(t; tqq) = Since = S. Albeveri
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hence C = C3 = S. Albeverio, A. Kos
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Furthermore, D. Brydges, A. Talarcz
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lim n→∞ Ω lim sup n→∞
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