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

K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580 563
Similarly, for X = D(eα, ¯cj ) − D(cj , ēα) ∈ k,wehave
and
Hence,
From this we obtain
Summarizing, we find on Vj,0,
Xa = D(eα, ¯cj ) − D(cj , ēα) a = tj eα,
X 2
a = X(tjeα) = tj D(eα, ¯cj ) − D(cj , ēα) eα =−tjcj .
∂ 2
4 f =
∂zα∂ ¯zβ
∂tj eα∂tj eαf(a)+ ∂−tj cj f(a)= 0.
∂2 ∂x2 α
f = 1 ∂
F.
tj ∂tj
0 if α = β,
∂ 2
∂x 2 α
+ ∂2
∂y2
1 ∂
f = 2 F if α = β.
α
tj ∂tj
Furthermore,
D (1) 2 (1) 2
b(a,ā)eα, ēα = D 1 − tj eα, ēα = 1 − tj D(eα, ēα) (1) .
Hence,
r
j=1 eα,eβ∈Vj,0
=
= 2
r
j=1 eα∈Vj,0
= 2b
D (1) ∂
b(a,ā)eα, ēβ
2
∂zα∂ ¯zβ
2
1 − tj D(eα, ēα) (1) 2 1
r 2 1 ∂F
1 − tj tj ∂tj
j=1
eα∈Vj,0
tj
f(a)
∂F
∂tj
D(eα, ēα) (1)
r 2 1 ∂F
1 − tj D(cj , ¯cj )
tj ∂tj
(1) ,
j=1
since we already proved in Lemma 5.1, that
D(eα, ēα) (1) = bD(cj ,cj ) (1) .
This finishes the proof. ✷
eα∈Vj,0