ExampleConstruction: Literature-sourcedVerification: 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_p range of a power weight

Example

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

Fix $1 < p < \infty$ and $α \in R$ , and let $w (x) : = ∣ x ∣^{α}$ on $R^{n}$ (the value at the origin being assigned arbitrarily, say $w (0) : = 0$ ). Then $w$ is a weight exactly when $α > - n$ , and for such $α$ one has $w \in A_{p}$ if and only if $- n < α < n (p - 1) .$ In that open range the $A_{p}$ characteristic is finite and bounded in terms of $n, p, α$ ; when $α \geq n (p - 1)$ the reciprocal-power average diverges on cubes containing the origin; when $α \leq - n$ , the function is not a weight. Both thresholds are local integrability conditions at the origin. Moreover the associated measure is doubling for every $α > - n$ .

Facts & Assumptions

Given: Countable Choice; $n \geq 1$ , $1 < p < \infty$ , $α \in R$ , and $w = ∣ x ∣^{α}$ .

[F1]

$w$ is a weight iff it is Lebesgue measurable, locally integrable and positive and finite a.e.; the $A_{p}$ characteristic is $\sup_{Q} ⟨ w ⟩_{Q} ⟨ w^{- 1 / (p - 1)} ⟩_{Q}^{p - 1}$ over cubes (Weights, their associated measures, and the spaces L^p(w), Muckenhoupt A_p and A_1 weights).

[F2]

Polar coordinates: $\int_{R^{n}} g (x) d x = \int_{0}^{\infty} \int_{S^{n - 1}} g (r θ) r^{n - 1} d θ d r$ , so $\int_{B (0, R)} ∣ x ∣^{a} d x = ∣ S^{n - 1} ∣ R^{n + a} / (n + a)$ for $a > - n$ and $+ \infty$ for $a \leq - n$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Continuity and derivatives of positive-base real powers, Comparison tests for improper integrals, Dominated convergence, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

[F3]

For $a > - n$ , $∣ x_{0} ∣ \leq 3 R$ implies $B (x_{0}, R) \subseteq B (0, 4 R)$ , so [F2] bounds its $∣ x ∣^{a}$ integral above by $C_{n, a} R^{n + a}$ . If $a < 0$ , then $∣ x ∣^{a} \geq (4 R)^{a}$ a.e. on this ball, giving a positive lower bound of that order. If $a \geq 0$ , remove $B (0, R / 2)$ ; the remainder has measure at least $(1 - 2^{- n}) ∣ B (x_{0}, R) ∣$ and $∣ x ∣^{a} \geq (R / 2)^{a}$ there, again giving a positive lower bound. For $∣ x_{0} ∣ \geq 3 R$ , $(2 / 3) ∣ x_{0} ∣ \leq ∣ x ∣ \leq (4 / 3) ∣ x_{0} ∣$ on the ball, so its integral is comparable to $∣ x_{0} ∣^{a} R^{n}$ . Volume scaling is For a nonzero real $c$ , dilation by $c$ multiplies Lebesgue outer measure by $∣ c ∣^{n}$ , and reflection in the origin preserves it, and ball/cube characteristic equivalence is Ball and cube maximal functions are pointwise comparable.

Verification

technique · direct

1.1F1F2F3givenalgebra

If $α \leq - n$ , then $\int_{B (0, 1)} ∣ x ∣^{α} d x = + \infty$ by [F2], so $w$ is not locally integrable and hence not a weight. If $α > - n$ , then $w$ is locally integrable (apply [F2] on each ball, using [F3] to compare $\int_{B (x_{0}, R)} ∣ x ∣^{α}$ with the radial integral), it is positive and finite off the origin, and it is assigned the value $0$ at the single point $0$ of measure zero; hence $w$ is a weight.

1.2F3givenalgebra

Let $α > - n$ and let $B = B (x_{0}, R)$ be a ball with $∣ x_{0} ∣ \geq 3 R$ . On $B$ one has $(2 / 3) ∣ x_{0} ∣ \leq ∣ x ∣ \leq (4 / 3) ∣ x_{0} ∣$ , so both $\int_{B} ∣ x ∣^{α} d x$ and $\int_{B} ∣ x ∣^{- α / (p - 1)} d x$ are comparable to $∣ x_{0} ∣^{α} R^{n}$ and $∣ x_{0} ∣^{- α / (p - 1)} R^{n}$ respectively, and the defining product is bounded by a constant depending only on $n, α, p$ .

2.1F3step 1.2givenalgebra

Let $α > - n$ and let $B = B (x_{0}, R)$ with $∣ x_{0} ∣ < 3 R$ . By [F3] the two integrals $\int_{B} ∣ x ∣^{α} d x$ and $\int_{B} ∣ x ∣^{- α / (p - 1)} d x$ are comparable to $R^{n + α}$ and $R^{n - α / (p - 1)}$ provided both exponents exceed $- n$ ; the normalized product $⟨ w ⟩_{B} ⟨ w^{- 1 / (p - 1)} ⟩_{B}^{p - 1}$ is then comparable to $R^{α} R^{- α / (p - 1) \cdot (p - 1)} = 1$ , uniformly in $B$ . If $- α / (p - 1) \leq - n$ , that is $α \geq n (p - 1)$ , the second exponent does not exceed $- n$ and the corresponding integral over $B (0, R)$ diverges, so the product is $+ \infty$ on the ball $B (0, R)$ .

3.1F2F3step 1.1step 1.2step 2.1givenalgebra∎

Steps 1.1–2.1 give the stated weight and $A_{p}$ ranges for ball averages; the ball/cube comparison [F3] gives the same result for the defined cube characteristic. For doubling, if $∣ x_{0} ∣ < 3 R$ , the integral on $B (x_{0}, 2 R)$ is bounded above by the radial integral on $B (0, 5 R)$ , of order $R^{n + α}$ , while [F3] bounds the integral on $B (x_{0}, R)$ below by a positive multiple of that order. If $∣ x_{0} ∣ \geq 3 R$ , both balls have $∣ x ∣$ comparable to $∣ x_{0} ∣$ (on the larger ball, $∣ x_{0} ∣ / 3 \leq ∣ x ∣ \leq 5 ∣ x_{0} ∣ / 3$ ); their integrals are therefore comparable up to a fixed constant. Hence $w (B (x_{0}, 2 R)) \leq C_{n, α} w (B (x_{0}, R))$ for every $α > - n$ .

Depends on

Used by

A power weight fails at both Aₚ endpoints Counterexample
The A₁ range of a power weight Example
Weighted norm of an interval indicator Example

Dependency tree · two levels

87 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)