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

D. Brydges, A. Talarczyk / Journal of Functional Analysis 236 (2006) 682–711 709
By continuity of ˜hm we have
lim
n→∞ h1
∗···∗hL−n(x) = E ˜hm
−(m+1)
x − L Y = h(x),
where the last equality follows by (5.8).
In a very similar way, possibly choosing larger m, we can show that also the derivatives of
h1 ∗···∗hn converge to the derivatives of h and are bounded. From this we obtain that for any
ϕ ∈ D(Rd ) we have (5.4) with Aϕ = h ∗ ϕ. Moreover, since for each n we have AL−n 1,
(5.4) holds for any ϕ ∈ H+(Rd ) and A 1.
To prove the other part of the theorem, let us denote
G−k(f, g) = G(f, g) − G A ′
L−k f,A ′
L−kf d
, f,g∈H−R , (5.9)
and
˜G−n(f, g) = G(f, g) − G A ′
1 ...A ′ ′
L−nf,A 1 ...A ′ L−ng d
, f,g∈H−R . (5.10)
Using (5.9) we can write
˜G−n(f, g) = G−n(f, g) +
n
k=1
′
G−(n−k) A
L−(n−k+1) ...A ′ ′
L−nf,A L−(n−k+1) ...A ′ L−ng . (5.11)
From Theorem 2.8 it follows that G−k is positive-definite for each k, hence so is ˜G−n.Fromthe
same theorem we have that range G−k is 2DL −k and we also know that
diam supp A ′
L −kϕ = diam supp ϕ + 2DL −k .
Combining these two facts with (5.11) we obtain that ˜G−n has range smaller than 2D +
2DL/(L − 1). By (5.4)
G(f, g) − G(A ′ f,A ′ G) = lim ˜G−n(f, g).
n→∞
Hence G(·,·) − G(A ′ ·,A ′ ·) is positive-definite and of range 2D + 2DL/(L − 1). ✷
Proof of Lemma 5.1. By (4.2) we have
ˆh(y) =
1
e
|x|2
ix·y
|B||x| d
∂B
σ(dz) 2x · z −|x| 2 1 2x·z|x| 2.
Function h is symmetric, hence ˆh is real, so we can replace e ix·y by cos(x · y). We change the
order of integration and substitute x = rw where r =|x| and w = x/|x| to obtain