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A_p weights are doubling

Statement

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

Let $1 \leq p < \infty$ and $w \in A_{p}$ (Muckenhoupt A_p and A_1 weights). Then the measure $w d λ$ is doubling: for every axis-parallel cube $Q$ and every $λ > 1$ , $w (λ Q) \leq λ^{n p} K_{p} w (Q),$ and for every ball $B (x, r)$ , $w (B (x, 2 r)) \leq (4 \sqrt{n})^{n p} K_{p} w (B (x, r)) .$ All constants depend only on $n$ , $p$ and $K_{p}$ , never on the particular cube, ball, or point.

Here $K_{p} = [w]_{A_{p}}$ for $p > 1$ , and $K_{1} = \sup_{Q} ⟨ w ⟩_{Q} / ({ess inf}_{Q} w)$ ; by The two defining forms of A_1 agree, $1 \leq K_{1} \leq c_{n} [w]_{A_{1}}$ . Thus the constants remain controlled by the stated $A_{p}$ data, including the ball-normalized endpoint characteristic.

Facts & Assumptions

Given: Countable Choice; $1 \leq p < \infty$ , $w \in A_{p}$ , a cube $Q$ , $λ > 1$ , and a ball $B (x, r)$ .

[F1]

For $w \in A_{p}$ , every cube satisfies $0 < w (Q) < \infty$ and $λ Q$ is an axis-parallel cube with $∣ λ Q ∣ = λ^{n} ∣ Q ∣$ (Axis-parallel cubes, their averages, and cube maximal functions, Muckenhoupt A_p and A_1 weights).

[F2]

For $w \in A_{p}$ , every cube $R$ and every nonnegative measurable $f$ obey $⟨ f ⟩_{R} \leq K_{p}^{1 / p} (w (R)^{- 1} \int_{R} f^{p} w)^{1 / p}$ for $p > 1$ ; for $p = 1$ , $w \geq ⟨ w ⟩_{R} / K_{1}$ a.e. gives the same bound with $K_{1}$ . (Weighted average comparison and the density-to-mass estimate for A_p weights, The two defining forms of A_1 agree).

[F3]

For every ball $B (x, r)$ one has $Q (x, r / \sqrt{n}) \subseteq B (x, r) \subseteq B (x, 2 r) \subseteq Q (x, 4 r)$ , and $Q (x, 4 r) = (4 \sqrt{n}) Q (x, r / \sqrt{n})$ (Ball and cube maximal functions are pointwise comparable, Axis-parallel cubes, their averages, and cube maximal functions), while $w d λ$ is a measure, so $A \subseteq B$ implies $w (A) \leq w (B)$ (Weights, their associated measures, and the spaces L^p(w)).

Proof

technique · direct

1.1F1F2givenalgebra

Apply [F2] on the cube $λ Q$ to $f : = 1_{Q}$ : then $⟨ f ⟩_{λ Q} = ∣ Q ∣ / ∣ λ Q ∣ = λ^{- n}$ by [F1] and $\int_{λ Q} f^{p} w d λ = w (Q)$ , so $λ^{- n} \leq K_{p}^{1 / p} (w (Q) / w (λ Q))^{1 / p}$ . Raising to the $p$ -th power and rearranging using $0 < w (Q), w (λ Q) < \infty$ gives $w (λ Q) \leq λ^{n p} K_{p} w (Q)$ , which is the cube assertion.

2.1F1F3step 1.1givenalgebra

For the ball assertion apply step 1.1 to the cube $Q_{0} : = Q (x, r / \sqrt{n})$ and the dilation factor $λ_{0} : = 4 \sqrt{n} > 1$ : by [F3], $B (x, 2 r) \subseteq Q (x, 4 r) = λ_{0} Q_{0}$ , and $Q_{0} \subseteq B (x, r)$ , so $w (B (x, 2 r)) \leq w (λ_{0} Q_{0}) \leq λ_{0}^{n p} K_{p} w (Q_{0}) \leq (4 \sqrt{n})^{n p} K_{p} w (B (x, r))$ .

3.1step 1.1step 2.1∎

Step 1.1 is the cube form with constant $λ^{n p} K_{p}$ and step 2.1 the ball form with constant $(4 \sqrt{n})^{n p} K_{p}$ ; both depend only on $n, p, K_{p}$ and the stated dilation, and no property of $Q$ , $x$ or $r$ entered otherwise. This is the asserted doubling property.

Depends on

Used by

The Muckenhoupt Aᵢnfinity class Definition
The weighted maximal function of a doubling weight Definition
The A₁ range of a power weight Example
The Aₚ range of a power weight Example
The Hardy-Littlewood maximal operator characterises Aₚ Theorem

Dependency tree · two levels

53 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)