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

S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 649
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
Aqn − xpqDpn +
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