LemmaStatement: 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 two defining forms of A_1 agree

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Let $w$ be a weight on $R^{n}$ (Weights, their associated measures, and the spaces L^p(w)). Then the following are equivalent:

$M w \leq C w$ almost everywhere for some constant $C < \infty$ , where $M$ is the centred ball maximal function (The centered and uncentered Hardy-Littlewood maximal functions);
$\sup_{Q} ⟨ w ⟩_{Q} ({ess inf}_{Q} w)^{- 1} = : C^{'} < \infty$ , the supremum over axis-parallel cubes, where the essential infimum is defined in Muckenhoupt A_p and A_1 weights.

Moreover the least constants satisfy $C^{'} \leq C_{n} C$ and $C \leq C_{n} C^{'}$ for a dimensional constant $C_{n}$ , and the same equivalence holds with the uncentred maximal function $M^{*}$ in place of $M$ . In particular the cube-average/ essential-infimum condition may be used as an equivalent definition of $A_{1}$ with a characteristic changed only by a dimensional factor; a positive average divided by a zero essential infimum is interpreted as $+ \infty$ ; the class $A_{1}$ itself is the one of Muckenhoupt A_p and A_1 weights.

Facts & Assumptions

Given: Countable Choice; A weight $w$ , the centred and uncentred ball maximal functions $M$ and $M^{*}$ , and the cube averages $⟨ w ⟩_{Q}$ .

[F1]

$w > 0$ and $w < \infty$ Lebesgue-a.e., and for every cube $Q$ one has $0 < w (Q) < \infty$ (Weights, their associated measures, and the spaces L^p(w)).

[F2]

For every ball $B = B (x, r)$ there is an axis-parallel cube $Q \supseteq B$ with $∣ Q ∣ \leq C_{n} ∣ B ∣$ , and for every cube $Q$ there is a ball $B \supseteq Q$ with $∣ B ∣ \leq C_{n} ∣ Q ∣$ ; consequently, if $E \subseteq F$ and $∣ F ∣ \leq C_{n} ∣ E ∣$ , then $⟨ w ⟩_{E} \leq C_{n} ⟨ w ⟩_{F}$ by nonnegativity (Ball and cube maximal functions are pointwise comparable).

[F3]

For a nonnegative function $g$ the set where $g$ does not satisfy a pointwise inequality of the form $g \leq c$ a.e. is contained in a null set, and countable unions of null sets are null (Measure-null sets and almost-everywhere statements relative to a measure).

[F4]

$Q^{n}$ is countable and dense in $R^{n}$ , so cubes with rational centre and rational side length approximate any given cube from outside with volume comparable by a fixed factor ( $Q^{n}$ is a countable dense subset of $R^{n}$ , and rational open boxes form a countable basis).

Proof

technique · direct

1.1F1F2givenalgebra

Assume (1), with constant $C$ . For each cube $Q$ of side $ℓ$ and each $x \in Q$ , the centred ball $B (x, \sqrt{n} ℓ)$ contains $Q$ and has volume at most a dimensional multiple of $∣ Q ∣$ . Thus $⟨ w ⟩_{Q} \leq C_{n} M w (x) \leq C_{n} C w (x)$ for almost every $x \in Q$ . Taking the essential infimum and then the supremum in $Q$ gives $C^{'} \leq C_{n} C$ . If the hypothesis instead uses $M^{*}$ , the same estimate holds since $M \leq M^{*}$ .

1.2F1F2F3F4givenchoose

Assume (2). For each rational-centred, rational-sided cube $Q$ , the set $N (Q) = {x \in Q : ⟨ w ⟩_{Q} > C^{'} w (x)}$ is null. Their union $N$ is null by [F3, F4]. For $x \notin N$ and any ball $B ∋ x$ , choose a rational cube $Q \supseteq B$ with $∣ Q ∣ \leq C_{n} ∣ B ∣$ . Then $⟨ w ⟩_{B} \leq C_{n} ⟨ w ⟩_{Q} \leq C_{n} C^{'} w (x)$ . Taking the supremum over these balls gives $M^{*} w (x) \leq C_{n} C^{'} w (x)$ , hence also $M w (x) \leq C_{n} C^{'} w (x)$ .

2.1step 1.1step 1.2algebra∎

Steps 1.1 and 1.2 prove the equivalence together with the comparable bounds $C^{'} \leq C_{n} C$ and $C \leq C_{n} C^{'}$ for one and the same dimensional constant $C_{n}$ (renaming constants if necessary), and each direction was proved both for $M$ and for $M^{*}$ , so the centred and uncentred forms of condition (1) are equivalent to (2). Therefore the cube-average/essential-infimum condition defines the same class as $M w \leq C w$ a.e., with characteristic changed only by dimensional factors.

Depends on

Used by

Dependency tree · two levels

41 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)