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

Furthermore,
D. Brydges, A. Talarczyk / Journal of Functional Analysis 236 (2006) 682–711 697
1Vx ¯B(I
0 if supp ϕ ⊂ Vx,
− pVx )ϕ =
0 if supp ϕ ⊂ Rd \ V x
where we used (3.2) in the second case, so that
A ϕ+ =¯BA ϕL2 |V | −1
dx 1Vx ¯B(I − pVx )ϕ L2 =|V | −1
dx 1Vx ¯B(I − pVx )ϕ L2 X(ϕ)
|V | −1 X(ϕ) ϕ+,
where X(ϕ) is the set of x such that 1Vx ¯B(I − pVx∩Λ)ϕ = 0. By the previous equation X(ϕ) is
contained in the set of x such that Vx intersects both supp ϕ and the complement of supp ϕ. If
x ∈ X(ϕ), then the smallest ball that covers Vx neither contains nor is disjoint from the smallest
ball that covers supp ϕ. Therefore the distance between the centres of these balls lies in the
interval [R − Dϕ,R+ Dϕ] and there are constants C1,C depending on U such that
|V | −1
X(ϕ) C1R −d (R + Dϕ) d − (R − Dϕ) d C Dϕ
. ✷
R
Proof of Proposition 2.9. (1) The first item follows immediately by induction using Theorem
2.8 with U replaced by Uj ,j = 1, 2,...,n.
(2) Given L, letU0 have diameter D 1 2 (1 − L−1 ). Recall from just before Proposition 2.9
that f1 = f and, for j 2, fj = A ′
j−1 fj−1. By induction using Lemma 3.8, for j 2,
j−1
supp fj ⊂ y: dist(y, supp f) D
We find, following the proof of Theorem 2.8, that
Gj (f, f ) = 2 fj ,T ′
j fj
−,Λ − T ′
j fj ,T ′
j fj
−,Λ .
k=1
L k
. (3.6)
By Lemma 3.10, Gj (f, g) = 0 when the supports of fj and the analogously defined gj are
separated by at least 2Lj D. Therefore, Gj has range 2D j k=1 Lk Lj .
(3) Recall that L ϕ = B ′ Bϕ for ϕ ∈ D(Λ) and that L is the Riesz isometry. The operators
Aj are self-adjoint operators on H+(Λ). It easily follows that A ′
j = LAjL −1 . Therefore
A ′ ′
n ...A 1f =An ...A1ϕ+ =ϕn+1+, (3.7)
−,Λ
where we have defined ϕj inductively by setting ϕ1 = ϕ and, for j>1, ϕj = Aj−1ϕj−1. Since
(3.6) says that the diameter of the support of fj is less than diam supp f + L j−1 and since
Lemma 3.8 allows us to make exactly the same induction for ϕj , the diameter of the support of
ϕj is bounded by diam supp ϕ + L j−1 which, for j − 1 p, is less than 2L j−1 because p 0is