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

678 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681
Remark C.2. In fact ∂I k+1
k+1 (λ)/∂λp > 0, 2 p k, forλ = (λr) k+1
r=1 ∈ Rk+1 , λr 0, 1 r
k + 1, since by (38) we have ∂Gk(λ)/∂λp = A p p(C(λ ]p[ )) > 0.
Let us suppose that I k k (0) = 0, i.e.
0 = I k k (0) = M12...k−1
12...k−1
For k = 3, k = 4 and k = 5wehave
c 2 1k−1
c11
+ (M12...k−3k−2
12...k−3k−1 )2
M 12...k−3
12...k−3 M12...k−2
12...k−2
I 3 3 (0) = M12 12
I 4 4 (0) = M123 123
I 5 5 (0) = M1234 1234
c 2 14
c11
c 2 13
c 2 12
c11
c11
+ (M12
14 )2
c11M 12
12
We prove that I k+1
k+1 (0) = 0. Indeed, we get
I k+1
c2 1k
k+1 (0) = M12...k 12...k
c11
+ (M12
1k−1 )2
c11M 12
12
+ (M123
12k−1 )2
M 12
12 M123
123
+ M12...k−1
12...k−1
M 12...k−2
− ck−1k−1
12...k−2
+ M12
12
− c22 = 0,
c11
+ (M12
13 )2
c11M 12
12
+ (M123
124 )2
+ (M12
1k )2
c11M 12
12
+ (M12...k−2k−1
12...k−2k ) 2
M 12...k−2
12...k−2 M12...k−1
12...k−1
+ M123
123
M12
− c33 ,
12
M 12
12 M123
123
+···
.
+ M1234
1234
M123 − c44
123
+ (M123
12k )2
M 12
12 M123
123
+···
+ M12...k
12...k
M 12...k−1
− ckk
12...k−1
Since by Corollary A.5 we have
A
k−1
k−1 (Ck) A k−1
k (Ck)
Ak k−1 (Ck) Ak k (Ck)
= A∅ k−1k
∅ (Ck)Ak−1k (Ck) or
A
k−1
k−1 (Ck) A k−1k
k−1k (Ck)
= A k k−1 (Ck) 2 ,
A ∅ ∅ (Ck) A k k (Ck)
we conclude that
M
12...k−1
12...k−1 (Ck) M 12...k−2
12...k−2 (Ck)
M12...k 12...k (Ck) M 12...k−2k
12...k−2k (Ck)
= M 12...k−2k−1
12...k−2k (Ck) 2 .
Hence
(M 12...k−2k−1
12...k−2k (Ck)) 2
M 12...k−2 12...k−1
12...k−2 (Ck)M12...k−1 12...k (Ck)
M 12...k−1
12...k−1
(Ck) + M12...k
.
.
12...k−2k (Ck)
M12...k−2k
=
(Ck) M 12...k−2
,
(Ck)
12...k−2