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Power decay implies membership in some A_p

Statement

Assume the Axiom of Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). Let $w$ be a weight on $R^{n}$ (Weights, their associated measures, and the spaces L^p(w)) and suppose there are constants $C, δ > 0$ with $\frac{w (E)}{w (Q)} \leq C (\frac{∣ E ∣}{∣ Q ∣})^{δ}$ for every axis-parallel cube $Q$ and every measurable $E \subseteq Q$ . Then $w \in A_{p}$ (Muckenhoupt A_p and A_1 weights) for $p = 1 + 1 / γ < \infty$ , where $γ > 0$ and the resulting bound on $[w]_{A_{p}}$ depend only on $n, C, δ$ ; consequently $w \in A_{\infty}$ (The Muckenhoupt A_infinity class).

Facts & Assumptions

Given: Dependent Choice, a weight $w$ , constants $C, δ > 0$ with the displayed power decay, and a cube $Q$ .

[F1]

$w d λ$ is a locally finite measure with $0 < w (Q) < \infty$ for every cube $Q$ ; subsets have smaller measure, and $w (\emptyset) = 0$ (Weights, their associated measures, and the spaces L^p(w)).

[F2]

Power decay implies that $w d λ$ is doubling, with a doubling constant depending only on $n, C, δ$ (Power decay implies doubling).

[F3]

Reverse Hölder from a distribution estimate: if $μ$ is a doubling measure of the form $v d λ$ and $h \geq 0$ is measurable with $h v \in L_{l o c}^{1} (λ)$ and with $μ (S) \leq α μ (Q) \Rightarrow \int_{S} h d μ \leq β \int_{Q} h d μ$ for some $0 < α, β < 1$ and every cube $Q$ and measurable $S \subseteq Q$ , then there are $q > 1$ and $c < \infty$ , depending only on $n$ , the doubling constant of $μ$ , $α$ and $β$ , with $(μ (Q)^{- 1} \int_{Q} h^{q} d μ)^{1 / q} \leq c μ (Q)^{- 1} \int_{Q} h d μ$ for every cube $Q$ (Reverse Holder from a distribution estimate for a doubling weight, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F4]

For real exponents, $1 - q = - (q - 1)$ and $p = q / (q - 1) = 1 + 1 / (q - 1)$ satisfies $p - 1 = 1 / (q - 1)$ and $p^{'} = q$ (Real powers for positive bases, with the zero-base positive-exponent convention).

Proof

technique · direct

1.1F1givenalgebra

The density implication. Put $β : = 1 - \min {\frac{1}{2}, (2 C)^{- 1 / δ}} \in (0, 1)$ and let $S \subseteq Q$ be measurable with $w (S) \leq \frac{1}{2} w (Q)$ . Then $∣ S ∣ \leq β ∣ Q ∣$ : otherwise $∣ Q ∖ S ∣ < (1 - β) ∣ Q ∣ \leq (2 C)^{- 1 / δ} ∣ Q ∣$ , so power decay applied to the complement gives $w (Q ∖ S) < \frac{1}{2} w (Q)$ , whence $w (S) > \frac{1}{2} w (Q)$ , a contradiction. Equivalently, writing $μ : = w d λ$ and $h : = w^{- 1}$ , the hypothesis of [F3] holds with $α = \frac{1}{2}$ and this $β$ : $μ (S) \leq α μ (Q)$ implies $\int_{S} h d μ = ∣ S ∣ \leq β ∣ Q ∣ = β \int_{Q} h d μ$ .

2.1F2F3step 1.1givenalgebra

Applying the reverse Hölder lemma. The measure $μ = w d λ$ is doubling by [F2], and $h = w^{- 1} \geq 0$ has $\int_{Q} h d μ = ∣ Q ∣ < \infty$ for every cube, so $h w \in L_{l o c}^{1} (λ)$ ; step 1.1 supplies the density implication with $α = \frac{1}{2}$ and $β < 1$ . By [F3] there are $q > 1$ and $c < \infty$ , depending only on $n$ , the doubling constant of $w$ , and hence only on $n, C, δ$ , such that, for every cube $Q$ , $(w (Q)^{- 1} \int_{Q} w^{1 - q} d λ)^{1 / q} \leq c ∣ Q ∣ / w (Q)$ , since $h^{q} d μ = w^{- q} \cdot w d λ = w^{1 - q} d λ$ and $\int_{Q} h d μ = ∣ Q ∣$ .

3.1F4step 2.1givenalgebra∎

From the reverse Hölder estimate to $A_{p}$ . Put $γ : = q - 1 > 0$ and $p : = q / (q - 1) = 1 + 1 / γ$ , so that $p - 1 = 1 / γ$ and $w^{- 1 / (p - 1)} = w^{- γ}$ ; write $I : = w (Q)^{- 1} \int_{Q} w^{- γ} d λ = ⟨ w^{- γ} ⟩_{Q} / ⟨ w ⟩_{Q}$ . Step 2.1 reads $I^{1 / q} \leq c ⟨ w ⟩_{Q}^{- 1}$ , hence $I^{1 / γ} \leq (c ⟨ w ⟩_{Q}^{- 1})^{q / γ} = c^{p} ⟨ w ⟩_{Q}^{- p}$ by [F4], since $q / γ = q / (q - 1) = p$ . Multiplying by $⟨ w ⟩_{Q}^{p}$ and using $⟨ w ⟩_{Q}^{p} I^{1 / γ} = ⟨ w ⟩_{Q}^{1 + 1 / γ} (⟨ w^{- γ} ⟩_{Q} / ⟨ w ⟩_{Q})^{1 / γ} = ⟨ w ⟩_{Q} ⟨ w^{- γ} ⟩_{Q}^{1 / γ}$ gives $⟨ w ⟩_{Q} ⟨ w^{- 1 / (p - 1)} ⟩_{Q}^{p - 1} = ⟨ w ⟩_{Q} ⟨ w^{- γ} ⟩_{Q}^{1 / γ} \leq c^{p}$ for every cube $Q$ . Therefore $[w]_{A_{p}} \leq c^{p} < \infty$ , that is, $w \in A_{p}$ , and $w \in A_{\infty}$ by the definition of the latter as the union of the finite-exponent classes.

Depends on

Used by

The Aᵢnfinity power-decay characterisation Theorem

Dependency tree · two levels

44 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)