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

S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 651
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