lemmaroutine

Square function bound for dilated vector kernels

Bounds square function of dilated vector kernels with only log⁡(1/δ)\log(1/\delta) loss
handles different dilates in the square function proof
Lemma·lem:square-dilates·§11 Stein's square function

Statement (paper)

Let HH be a Hilbert space, 2<p<∞2<p<\infty, 0<δ≤10<\delta\le1, and N=n+3N=n+3. Suppose K:Rn→HK:\RR^n\to H is continuously differentiable and ∥K(x)∥H+∥∇K(x)∥Hn≤C∗δ−N(1+∣x∣)−N. \|K(x)\|_H+\|\nabla K(x)\|_{H^n} \le C_*\delta^{-N}(1+|x|)^{-N}. Write S(k)f=2nkK(2k⋅)∗fS^{(k)}f=2^{nk}K(2^k\cdot)*f, and let BqB_q bound the norm of K∗K* from LqL^q to Lq(H)L^q(H), q=2,pq=2,p. For every finite sequence of Schwartz functions fkf_k, ∥(∑k∥S(k)fk∥H2)1/2∥p≲n,p(B2+C∗δ+[1+log⁡(1/δ)]1/2−1/pBp)∥(∑k∣fk∣2)1/2∥p. \left\|\left(\sum_k\|S^{(k)}f_k\|_H^2\right)^{1/2}\right\|_p \lesssim_{n,p} \left(B_2+C_*\delta+[1+\log(1/\delta)]^{1/2-1/p}B_p\right) \left\|\left(\sum_k|f_k|^2\right)^{1/2}\right\|_p .

In plain English

For a Hilbert-space-valued kernel KK with decay and gradient bounds of size C∗δ−N(1+∣x∣)−NC_*\delta^{-N}(1+|x|)^{-N}, N=n+3N=n+3, the vector-valued square function ∥(∑k∥S(k)fk∥H2)1/2∥p\|(\sum_k\|S^{(k)}f_k\|_H^2)^{1/2}\|_p for dilates S(k)f=2nkK(2k⋅)∗fS^{(k)}f=2^{nk}K(2^k\cdot)*f is bounded by a constant times ∥(∑k∣fk∣2)1/2∥p\|(\sum_k|f_k|^2)^{1/2}\|_p. The constant is B2+C∗δ+[1+log⁡(1/δ)]1/2−1/pBpB_2+C_*\delta+[1+\log(1/\delta)]^{1/2-1/p}B_p, where B2,BpB_2,B_p are the Lq→Lq(H)L^q\to L^q(H) norms of K∗K*, valid for 2<p<∞2<p<\infty and 0<δ≤10<\delta\le1.

Why it's here

This lemma lets one apply a different dilate of the same operator to each fkf_k and still control the result by the square function of the fkf_k, with only a logarithmic loss in δ\delta at LpL^p. It is the tool in Stein's square function argument that handles shell pieces at scale δ\delta and the smooth part away from the boundary, with explicit dependence on δ\delta tracked through the BpB_p term.
Classified routine:
A standard vector-valued Calderón–Zygmund / sharp-function interpolation argument for square functions of dilates, as in Seeger's and Carbery's work on Bochner–Riesz maximal functions. The explicit tracking of the log(1/δ) factor is bookkeeping on that known technique.

Uses

Nothing else in the paper

Used by

thmStein's square function bounded on LpclaimShell pieces sum to a convergent boundclaimSmooth part away from ball boundary is bounded

Sources

PaperOpen
Section
§11 Stein's square function
Page
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Label
lem:square-dilates
Line
sections/stein_square.tex:103
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