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

S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 677
To define the function I k+1
k+1 k+1
k+1 (λ) with Ik+1 Ik+1 (ˆλ) we have
I k+1
k+1
k+1
= fk+1Ak+1 Ck+1(ˆλ) − ˆλk+1A k+1
k+1 (Ck+1)
= fk + f k+1 A k+1
Ck+1(λ) − ˆλk+1A k+1
(75)
(76)
(54)
k+1
ˆλkA kk+1
kk+1 (Ck+1)
A kk+1
kk+1
(Ck+1(λ)) + e
ˆλkA kk+1
kk+1 (Ck+1)
A kk+1
kk+1
:= I k+1
k+1 (ˆλ),
(Ck+1(λ)) + e
k
r=1
k
r=1
Ψ rk
Ψ rk
0
A k+1
k+1
A k+1
k+1
k+1 (Ck+1) λ=ˆλ
Ck+1(λ) − ˆλk+1A k+1
Ck+1(λ) − ˆλk+1A k+1
where the function I k+1
pq
k+1 (λ) is defined by (see definition (54) of Ψ0 ):
r=2
k+1 (Ck+1)
λ=ˆλ
k+1 (Ck+1)
λ=ˆλ
I k+1
ˆλkM
k+1 (λ) =
12...k−1
12...k−1 (Ck+1)
M 12...k−1
12...k−1 (Ck+1(λ)) + c2 k (M 1k
+
c11
r=2
12...r−1r
12...r−1k
(Ck+1))
2
M 12...r−1
12...r−1
(Ck+1)M12...r 12...r (Ck+1(λ))
× M 12...k
12...k Ck+1(λ) − ˆλk+1M 12...k
12...k (Ck+1)
ˆλkM
=
12...k−1
12...k−1 (Ck+1)
M 12...k−1
12...k−1 (Ck+1(λ)) + c2 k−1
(M 1k
+
c11
12...r−1r
12...r−1k
(Ck+1))
2
M 12...r−1
12...r−1
(Ck+1)M12...r 12...r (Ck+1(λ))
× M 12...k
12...k Ck+1(λ) + (M12...k
12...k
(Ck+1))
2
M 12...k−1
12...k−1 (Ck+1) − ˆλk+1M 12...k
12...k (Ck+1).
Finally we have the following expression for I k+1
k+1 (λ) with corresponding positive constants ar,
2 r k − 1 (depending on k) and b1 ∈ R:
I k+1
k+1 (λ) =
=
k−1
a1 +
r=2
k−1
ar
a1 +
Gr(λ)
r=2
ar
M12...r 12...r (Ck+1(λ))
Gk(λ) + b1.
M 12...k
12...k
By (37) of Lemma A.7 we conclude that for λr 0, 2 r k, holds
∂I k+1
k+1 (λ)
∂λp
∂I k+1
k+1 (λ)
∂λk
=
∂Gk(λ)
= a1
∂λp
k−1
a1 +
r=2
k−1
+
r=2
ar
ar ∂Gk(λ)
0,
Gr(λ) ∂λk
∂
∂λp
Ck+1(λ) + b1
Gk(λ)
0, 2 p k. (78)
Gr(λ)