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

S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 653
(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