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

S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 659
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