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Kernel tail integrals of weighted L-p functions are finite

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $1 \leq p < \infty$ , $w \in A_{p}$ (Muckenhoupt A_p and A_1 weights), and let $k : R^{n} ∖ {0} \to C$ be measurable and satisfy the pointwise size bound $∣ k (y) ∣ \leq A_{1} ∣ y ∣^{- n}$ for $y \neq 0$ . Then for every $f \in L^{p} (w)$ , every $x \in R^{n}$ and every $ε > 0$ , $\int_{∣ y ∣ \geq ε} ∣ k (y) ∣ ∣ f (x - y) ∣ d y \leq C (n, p, [w]_{A_{p}}) A_{1} w (Q (x, ε))^{- 1 / p} ∥ f ∥_{L^{p} (w)},$ a finite bound depending only on the stated data and on the cube $Q (x, ε)$ . Consequently the truncated singular integrals $T_{ε} f$ and $T^{(ε, N)} f$ (Maximal truncated singular integrals) are defined at every point by absolutely convergent integrals, and they are Borel measurable functions of the centre $x$ .

Facts & Assumptions

Given: Countable Choice, $1 \leq p < \infty$ , $w \in A_{p}$ , the size bound $∣ k (y) ∣ \leq A_{1} ∣ y ∣^{- n}$ , $f \in L^{p} (w)$ , $x \in R^{n}$ and $ε > 0$ .

[F1]

Write $D_{p} = [w]_{A_{p}}^{1 / p}$ for $p > 1$ and $D_{1} = c_{n} [w]_{A_{1}}$ . Weighted average comparison: for every cube $Q$ and nonnegative measurable $g$ , $⟨ g ⟩_{Q} \leq D_{p} (w (Q)^{- 1} \int_{Q} g^{p} w)^{1 / p}$ , so $\int_{Q} ∣ f ∣ \leq D_{p} ∣ Q ∣ w (Q)^{- 1 / p} ∥ f ∥_{L^{p} (w)}$ (Weighted average comparison and the density-to-mass estimate for A_p weights).

[F2]

Power decay: since $w \in A_{\infty}$ (The Muckenhoupt A_infinity class), there are $C, δ > 0$ , depending only on $n$ and the data of a witnessing exponent, with $w (E) / w (Q) \leq C (∣ E ∣ / ∣ Q ∣)^{δ}$ for measurable $E \subseteq Q$ (A_infinity weights satisfy power decay).

[F3]

$w d λ$ is a locally finite measure and $0 < w (Q) < \infty$ for every cube $Q$ (Weights, their associated measures, and the spaces L^p(w), Muckenhoupt A_p and A_1 weights).

[F4]

Monotone convergence passes through increasing nonnegative sums (Monotone convergence for the integral). For a jointly measurable nonnegative integrand, the integral over a product space may be computed by iterated integrals (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product), and $∥ f ∥_{L^{p} (w)}^{p} = \int ∣ f ∣^{p} w d λ$ (Weights, their associated measures, and the spaces L^p(w)).

[F5]

Translations are norm-continuous in complex $L^{1}$ (Complex translation, convolution, approximate identities, and mollification); dominated convergence applies under an integrable majorant (Dominated convergence).

Proof

technique · direct

1.1F4givenalgebra

Cover ${∣ y ∣ \geq ε}$ by the annuli $A_{j} = {2^{j} ε \leq ∣ y ∣ < 2^{j + 1} ε}$ , $j \geq 0$ , which are pairwise disjoint with union ${∣ y ∣ \geq ε}$ and each contained in the ball $B (x, 2^{j + 2} ε)$ after the substitution $y \mapsto x - y$ . Hence $\int_{∣ y ∣ \geq ε} ∣ k (y) ∣ ∣ f (x - y) ∣ d y \leq \sum_{j \geq 0} (2^{j} ε)^{- n} A_{1} \int_{A_{j}} ∣ f (x - y) ∣ d y \leq \sum_{j \geq 0} (2^{j} ε)^{- n} A_{1} \int_{Q (x, 2^{j + 2} ε)} ∣ f ∣$ by monotone convergence and monotonicity of the integral.

1.2F1F2F3givenalgebra

For each $j \geq 0$ put $Q_{j} : = Q (x, 2^{j + 2} ε)$ and $R : = Q (x, ε)$ . By [F1], $\int_{Q_{j}} ∣ f ∣ \leq D_{p} ∣ Q_{j} ∣ w (Q_{j})^{- 1 / p} ∥ f ∥_{L^{p} (w)}$ , while $(2^{j} ε)^{- n} ∣ Q_{j} ∣ = (2^{j} ε)^{- n} (2^{j + 3} ε)^{n} = 2^{3 n}$ is a dimensional constant. By [F2] applied to $R \subseteq Q_{j}$ , $w (R) / w (Q_{j}) \leq C (∣ R ∣ / ∣ Q_{j} ∣)^{δ} = C 2^{- (j + 2) n δ}$ , so $w (Q_{j})^{- 1 / p} \leq C^{1 / p} 2^{- (j + 2) n δ / p} w (R)^{- 1 / p}$ .

2.1F2step 1.1step 1.2givenalgebra

Substituting the bounds of step 1.2 into step 1.1 gives $\int_{∣ y ∣ \geq ε} ∣ k (y) ∣ ∣ f (x - y) ∣ d y \leq 2^{3 n} D_{p} C^{1 / p} A_{1} ∥ f ∥_{L^{p} (w)} w (R)^{- 1 / p} \sum_{j \geq 0} 2^{- (j + 2) n δ / p}$ , and the geometric series converges because $δ > 0$ ; this is the asserted finite bound with $C (n, p, [w]_{A_{p}}) = 2^{3 n} D_{p} C^{1 / p} (1 - 2^{- n δ / p})^{- 1} 2^{- 2 n δ / p}$ .

3.1F5step 2.1givenalgebra∎

The estimate proves absolute convergence of every truncation. For fixed $0 < ε < N$ , the annular kernel is bounded and compactly supported. Near a fixed $x_{0}$ , truncate $f$ to a bounded ball containing all arguments $x - y$ under consideration, obtaining an $L^{1}$ function $f_{0}$ . Then $∣ T^{(ε, N)} f (x + h) - T^{(ε, N)} f (x) ∣ \leq A_{1} ε^{- n} ∥ f_{0} (\cdot + h) - f_{0} ∥_{1} \to 0$ by [F5]. Thus each finite truncation is continuous and Borel. Dominated convergence gives $T_{ε} f = \lim_{N \to \infty} T^{(ε, N)} f$ , which is Borel. Continuity of the defining integrals in the cutoff radii follows from absolute integrability and null spherical boundaries, so rational cutoffs suffice in both maximal suprema; these maximal functions are Borel as well.

Depends on

Used by

Weighted L-p bounds for standard Calderon-Zygmund maximal truncations Theorem

Dependency tree · two levels

81 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (Springer GTM 249, 2014) (standard reference, not scraped)
Juha Kinnunen, Harmonic Analysis (Aalto University lecture notes) (standard reference, not scraped)