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

688 D. Brydges, A. Talarczyk / Journal of Functional Analysis 236 (2006) 682–711
Now we give some results explaining in more detail the construction of the decomposition of
Theorem 2.6.
For U any convex bounded open subset of Λ, letpU be the orthogonal projection in H+(Λ)
onto the subspace H+(U). In particular pU = 0forU =∅.Weset
Ux = (x + U)∩ Λ.
Lemma 2.7. For each ϕ ∈ H+(Λ) there exists a unique vector Tϕin H+(Λ) such that
(T ϕ, ψ)+ = 1
|U|
The linear operator T : H+(Λ) ↦→ H+(Λ) is a contraction.
dx(pUx ϕ,ψ)+ for all ψ ∈ H+(Λ). (2.6)
The main ingredient of the decomposition is the following theorem allowing to “cut out” from
G a bilinear form which is positive-definite and of finite range.
Theorem 2.8. Let U be a convex, bounded open subset of Λ.Forf ∈ H−(Λ), define
Then the bilinear form G1 such that
is:
(1) positive-definite;
(2) finite range with range 2diamU.
A ′ U f = f − T ′ f. (2.7)
G(f, f ) = G1(f, f ) + G A ′ U f,A ′
U f
We construct the finite range decomposition by an iterated application of Theorem 2.8. Let
U0 be a convex open bounded set containing the origin and for j = 1, 2,...let Uj be a sequence
of domains obtained by scaling U,
For each domain construct T ′
j
Uj = x: L −j x ∈ U .
(2.8)
, A ′
j , ˜Gj by replacing U by Uj in the construction of G1 given
above. Set f1 = f and for j 2, set fj = A ′
j−1 fj−1. Define bilinear forms Gj for j 1by
Gj (f, f ) = ˜Gj (fj ,fj ).
Proposition 2.9.
(1) G(f, f ) = n j=1 Gj (f, f ) +A ′
n ...A ′
1f 2 −,Λ .
(2) Given L>1, let the diameter of U0 be chosen less than 1
2 (1 − L−1 ). Then for all j 1 the
range of Gj is less than Lj .