TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The A_infinity power-decay characterisation

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)). Then the following are equivalent:

$w \in A_{\infty}$ (The Muckenhoupt A_infinity class), that is, $w \in A_{p}$ (Muckenhoupt A_p and A_1 weights) for some $1 \leq p < \infty$ ;
there are $C, δ > 0$ with $w (E) / w (Q) \leq C (∣ E ∣ / ∣ Q ∣)^{δ}$ for every axis-parallel cube $Q$ and every measurable $E \subseteq Q$ ;
$w$ satisfies a reverse Hölder inequality: there are $γ > 0$ and $C < \infty$ with $(⟨ w^{1 + γ} ⟩_{Q})^{1 / (1 + γ)} \leq C ⟨ w ⟩_{Q}$ for every axis-parallel cube $Q$ .

All constants in each condition depend only on $n$ and on the constants appearing in the assumed condition.

Facts & Assumptions

Given: Dependent Choice; a weight $w$ ; the three conditions (i), (ii), (iii) of the statement.

[F1]

(i) implies (ii): for $w \in A_{\infty}$ there are $C, δ > 0$ , depending only on $n$ , a witnessing exponent $p$ and $[w]_{A_{p}}$ , with $w (E) / w (Q) \leq C (∣ E ∣ / ∣ Q ∣)^{δ}$ for every cube $Q$ and measurable $E \subseteq Q$ (A_infinity weights satisfy power decay).

[F2]

(ii) implies (i): power decay with constants $C, δ$ forces $w \in A_{p}$ for $p = 1 + 1 / γ < \infty$ with $γ > 0$ and the bound on $[w]_{A_{p}}$ depending only on $n, C, δ$ , hence $w \in A_{\infty}$ (Power decay implies membership in some A_p).

[F3]

(i) implies (iii): for $1 \leq p < \infty$ and $w \in A_{p}$ there are $γ > 0$ and $C < \infty$ , depending only on $n$ , $p$ and $[w]_{A_{p}}$ , with $(⟨ w^{1 + γ} ⟩_{Q})^{1 / (1 + γ)} \leq C ⟨ w ⟩_{Q}$ for every cube $Q$ (Reverse Holder self-improvement for A_p weights); this is applied to a witnessing exponent of $A_{\infty}$ .

[F4]

Hölder's inequality for the conjugate exponents $1 + γ$ and $(1 + γ) / γ$ : $\int_{E} w d λ \leq (\int_{Q} w^{1 + γ} d λ)^{1 / (1 + γ)} ∣ E ∣^{γ / (1 + γ)}$ for measurable $E \subseteq Q$ ; all cube averages are those of Muckenhoupt A_p and A_1 weights and $(1 + γ) / γ > 1$ is a real exponent (Holder's inequality for integrals, including the endpoint cases, Real powers for positive bases, with the zero-base positive-exponent convention).

Proof

technique · direct

1.1F1given

(i) implies (ii). If $w \in A_{\infty}$ , then $w \in A_{p}$ for a witnessing $1 \leq p < \infty$ , and [F1] supplies $C, δ > 0$ , depending only on $n$ , $p$ and $[w]_{A_{p}}$ , with $w (E) / w (Q) \leq C (∣ E ∣ / ∣ Q ∣)^{δ}$ for every cube $Q$ and measurable $E \subseteq Q$ .

1.2F2given

(ii) implies (i). If power decay holds with constants $C, δ$ , then [F2] gives $γ > 0$ and $p = 1 + 1 / γ < \infty$ with $w \in A_{p}$ and $[w]_{A_{p}}$ bounded in terms of $n, C, δ$ ; by the definition of $A_{\infty}$ as the union of the classes $A_{p}$ , $1 \leq p < \infty$ , this is (i).

1.3F3given

(i) implies (iii). Let $w \in A_{p}$ with witnessing exponent $p$ , which may be taken finite by (i). By [F3] there are $γ > 0$ , $C < \infty$ , depending only on $n$ , $p$ and $[w]_{A_{p}}$ , with $(⟨ w^{1 + γ} ⟩_{Q})^{1 / (1 + γ)} \leq C ⟨ w ⟩_{Q}$ for every cube $Q$ , which is condition (iii).

1.4F4givenalgebra

(iii) implies (ii). Suppose (iii) holds with $γ > 0$ and $C$ . For a cube $Q$ and measurable $E \subseteq Q$ , [F4] gives $\int_{E} w d λ \leq (\int_{Q} w^{1 + γ} d λ)^{1 / (1 + γ)} ∣ E ∣^{γ / (1 + γ)}$ ; substituting $\int_{Q} w^{1 + γ} d λ = ∣ Q ∣ ⟨ w^{1 + γ} ⟩_{Q} \leq C^{1 + γ} w (Q)^{1 + γ} ∣ Q ∣^{- γ}$ (the $(1 + γ)$ -th power of the reverse Hölder inequality, since $⟨ w ⟩_{Q} = w (Q) / ∣ Q ∣$ ) yields $w (E) \leq C w (Q) (∣ E ∣ / ∣ Q ∣)^{γ / (1 + γ)}$ , that is, (ii) with this same $C$ and $δ = γ / (1 + γ)$ .

2.1step 1.1step 1.2step 1.3step 1.4∎

The cycles (i) $\Rightarrow$ (ii) $\Rightarrow$ (i) of steps 1.1 and 1.2 and (i) $\Rightarrow$ (iii) $\Rightarrow$ (ii) of steps 1.3 and 1.4 exhibit each of the three conditions as equivalent to the others; the constants recorded in steps 1.1, 1.2, 1.3 and 1.4 depend only on $n$ and on the constants appearing in the assumed condition, as claimed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

54 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)