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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.

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The A_1 range of a power weight

Example

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

For $α > - n$ the power weight $w (x) = ∣ x ∣^{α}$ belongs to $A_{1}$ if and only if $- n < α \leq 0.$ For $- n < α \leq 0$ the function is radially nonincreasing and locally integrable and satisfies $M (∣ x ∣^{α}) \leq C_{n, α} ∣ x ∣^{α}$ almost everywhere; for $α > 0$ it is continuous with value $0$ at the origin and fails the cube-average/essential-infimum form of $A_{1}$ on cubes centred at the origin; for $α \leq - n$ it is not locally integrable. Thus the $A_{1}$ exponent interval is the interval obtained as the limit of the ranges at $p ↓ 1$ , and it is a strict subset of the doubling range $α > - n$ .

Facts & Assumptions

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

[F1]

For $α > - n$ , $w$ is a weight, and by The two defining forms of A_1 agree the condition $w \in A_{1}$ is equivalent to $M (∣ x ∣^{α}) \leq C ∣ x ∣^{α}$ a.e. for some $C < \infty$ (The A_p range of a power weight, Muckenhoupt A_p and A_1 weights, Weights, their associated measures, and the spaces L^p(w)).

[F2]

For $α > - n$ the radial integral over a ball is $\int_{B (0, ρ)} ∣ z ∣^{α} d z = ∣ S^{n - 1} ∣ ρ^{n + α} / (n + α)$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma), the ball $B (y, r) ∋ x$ with $∣ x ∣ \geq 4 r$ satisfies $∣ z ∣ \geq ∣ x ∣ / 2$ for all $z \in B (y, r)$ , and $B (y, r) \subseteq B (0, ∣ x ∣ + 2 r)$ whenever $∣ x ∣ < 4 r$ (elementary triangle inequality, Continuity and derivatives of positive-base real powers for the monotonicity of $t \mapsto t^{α}$ ).

Verification

technique · direct

1.1F1F2givenalgebra

Let $- n < α \leq 0$ and let $B (y, r) ∋ x \neq 0$ . If $∣ x ∣ \geq 4 r$ , then $∣ z - x ∣ < 2 r$ for $z \in B (y, r)$ , so $∣ z ∣ \geq ∣ x ∣ / 2$ and $∣ z ∣^{α} \leq 2^{- α} ∣ x ∣^{α}$ . If $∣ x ∣ < 4 r$ , then $B (y, r) \subseteq B (0, ∣ x ∣ + 2 r) \subseteq B (0, 6 r)$ ; [F2] bounds its integral by $C_{n, α} r^{n + α}$ and $r^{α} \leq 4^{- α} ∣ x ∣^{α}$ . Dividing by the ball volume gives an average bounded by $C_{n, α}^{'} ∣ x ∣^{α}$ in both cases. Taking the supremum gives the uncentred bound, hence also the centred bound and $A_{1}$ membership.

1.2F1givenalgebra

For $α > 0$ and a cube $Q$ centred at the origin, ${ess inf}_{Q} ∣ x ∣^{α} = 0$ because every positive threshold has a subball around the origin on which $∣ x ∣^{α}$ lies below it; that subball has positive Lebesgue measure, while $⟨ ∣ x ∣^{α} ⟩_{Q} > 0$ ; hence the cube-average/essential-infimum form of $A_{1}$ fails on $Q$ , and by [F1] the pointwise form fails as well.

2.1step 1.1step 1.2given∎

For $α \leq - n$ the function is not locally integrable, hence not a weight, by The A_p range of a power weight. Together with steps 1.1 and 1.2 this shows that $A_{1}$ membership holds exactly for $- n < α \leq 0$ , while the doubling range for the measure $∣ x ∣^{α} d x$ is the strictly larger interval $α > - n$ (the direct doubling computation in The A_p range of a power weight, the final verification step).

Depends on

Used by

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Dependency tree · two levels

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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)