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Weighted good-lambda inequality for maximal truncations

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $w \in A_{\infty}$ (The Muckenhoupt A_infinity class), let $T$ be a standard-kernel Calderón–Zygmund operator as in the unweighted local estimate, let $λ > 0$ , and let $f \in L_{l o c}^{1} (R^{n})$ with the defining finiteness property and with ${T^{* *} f > λ}$ a proper open set. Then there are $γ_{0} > 0$ , $C < \infty$ and $δ^{'} > 0$ , depending only on $n$ , the Hölder exponent $δ$ , a witnessing finite exponent $p$ and $[w]_{A_{p}}$ and the constants $A_{1}, A_{2}^{'}, A_{3}, B$ , such that for every $0 < γ < γ_{0}$ , $w ({T^{* *} f > 2 λ} \cap {M f \leq γ λ}) \leq C γ^{δ^{'}} w ({T^{* *} f > λ}) .$ The same holds with $T^{*}$ throughout when its own level set ${T^{*} f > λ}$ is a proper open set.

Facts & Assumptions

Given: Countable Choice, $w \in A_{\infty}$ , the operator data $A_{1}, A_{2}^{'}, A_{3}, B, δ$ , the function $f$ and the height $λ$ .

[F1]

Unweighted local estimate: with $Ω_{λ} = {T^{* *} f > λ}$ decomposed into the Whitney cubes $Q_{j}$ , for $γ < γ_{0}^{u n w}$ the set $E_{j} : = Q_{j} \cap {T^{* *} f > 2 λ} \cap {M f \leq γ λ}$ satisfies $∣ E_{j} ∣ \leq C_{n} γ (A_{1} + A_{2}^{'} + A_{3} + B) ∣ Q_{j} ∣$ ; the cubes $Q_{j}$ are pairwise disjoint with union $Ω_{λ}$ (Unweighted local good-lambda estimate for maximal truncations).

[F2]

Power decay: since $w \in A_{\infty}$ , there are $C_{w}, η > 0$ , depending only on $n$ and the $A_{\infty}$ data, such that $w (E) / w (Q) \leq C_{w} (∣ E ∣ / ∣ Q ∣)^{η}$ for every cube $Q$ and measurable $E \subseteq Q$ (A_infinity weights satisfy power decay); in particular $0 < w (Q) < \infty$ for every cube.

[F3]

Lebesgue measure and $w d λ$ are countably additive on pairwise disjoint measurable sets (Countable additivity and continuity of finitely additive set functions).

Proof

technique · direct

1.1F2givenalgebra

Keep the Whitney decomposition of $Ω_{λ}$ supplied by [F1] and let $E_{j}$ be as there. Each $E_{j}$ is a measurable subset of $Q_{j}$ with $∣ E_{j} ∣ \leq (C_{n} γ (A_{1} + A_{2}^{'} + A_{3} + B)) ∣ Q_{j} ∣$ , so the power decay [F2] gives $w (E_{j}) \leq C_{w} (C_{n} γ (A_{1} + A_{2}^{'} + A_{3} + B))^{η} w (Q_{j})$ .

2.1F1F3step 1.1givenalgebra

Summing step 1.1 over the pairwise disjoint $Q_{j}$ , whose union is $Ω_{λ}$ by [F1], and using countable additivity [F3]: $w ({T^{* *} f > 2 λ} \cap {M f \leq γ λ}) = \sum_{j} w (E_{j}) \leq C_{w} C_{n}^{η} (A_{1} + A_{2}^{'} + A_{3} + B)^{η} γ^{η} \sum_{j} w (Q_{j}) = C_{w} C_{n}^{η} (A_{1} + A_{2}^{'} + A_{3} + B)^{η} γ^{η} w (Ω_{λ})$ .

3.1F1step 2.1givenalgebra∎

The display of step 2.1 is the claimed inequality with $δ^{'} = η$ and $C = C_{w} C_{n}^{η} (A_{1} + A_{2}^{'} + A_{3} + B)^{η}$ , valid for every $0 < γ < γ_{0}$ with $γ_{0} = γ_{0}^{u n w}$ from [F1]; the same argument applies with $T^{*}$ in place of $T^{* *}$ by applying the unweighted lemma to its own level set and Whitney decomposition.

Depends on

Used by

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

Dependency tree · two levels

32 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)