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Duality and nesting of the A_p classes

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:

For $1 < p < \infty$ , $w \in A_{p}$ if and only if its reciprocal power $w^{- 1 / (p - 1)}$ lies in $A_{p^{'}}$ , where $1 / p + 1 / p^{'} = 1$ , and then $[w^{- 1 / (p - 1)}]_{A_{p^{'}}} = [w]_{A_{p}}^{1 / (p - 1)}$ (Conjugate exponents, including the endpoint conventions).
The classes are nested: $A_{p} \subseteq A_{q}$ for $1 < p < q < \infty$ with $[w]_{A_{q}} \leq [w]_{A_{p}}$ , and $A_{1} \subseteq A_{q}$ for every $1 < q < \infty$ with $[w]_{A_{q}} \leq C_{n} [w]_{A_{1}}$ for a dimensional constant $C_{n}$ .

Facts & Assumptions

Given: Countable Choice; A weight $w$ and the characteristic constants $[w]_{A_{p}} = \sup_{Q} ⟨ w ⟩_{Q} ⟨ w^{- 1 / (p - 1)} ⟩_{Q}^{p - 1}$ of Muckenhoupt A_p and A_1 weights.

[F1]

The $A_{1}$ condition is equivalent to the cube-average/essential-infimum form: there is a dimensional constant $c_{n} \geq 1$ with $⟨ w ⟩_{Q} \leq c_{n} [w]_{A_{1}} {ess inf}_{Q} w$ for every cube $Q$ , and conversely the cube-average form with constant $C^{'}$ gives $[w]_{A_{1}} \leq c_{n} C^{'}$ (The two defining forms of A_1 agree).

[F2]

Hölder's inequality with conjugate exponents $r, r^{'}$ and the generalized form for finitely many factors hold for nonnegative measurable functions; in particular $⟨ g^{θ} ⟩_{Q} \leq ⟨ g ⟩_{Q}^{θ}$ for $0 < θ \leq 1$ and nonnegative measurable $g \in L_{l o c}^{1}$ (Holder's inequality for integrals, including the endpoint cases, Generalized Holder inequality puts products into $L^{r}$ ).

Proof

technique · direct

1.1F2givenalgebra

Set $v : = w^{- 1 / (p - 1)}$ . Since $p^{'} - 1 = 1 / (p - 1)$ , one has $v^{- 1 / (p^{'} - 1)} = v^{- (p - 1)} = w$ , and therefore $⟨ v ⟩_{Q} ⟨ v^{- 1 / (p^{'} - 1)} ⟩_{Q}^{p^{'} - 1} = ⟨ w^{- 1 / (p - 1)} ⟩_{Q} ⟨ w ⟩_{Q}^{1 / (p - 1)} = (⟨ w ⟩_{Q} ⟨ w^{- 1 / (p - 1)} ⟩_{Q}^{p - 1})^{1 / (p - 1)}$ for every cube $Q$ . Taking suprema, the two suprema are finite simultaneously and $[v]_{A_{p^{'}}} = [w]_{A_{p}}^{1 / (p - 1)}$ ; if $w \in A_{p}$ , finiteness of its product and positivity of $⟨ w ⟩_{Q}$ imply local integrability of $v$ , so $v$ is a weight. Conversely, if $v \in A_{p^{'}}$ , the displayed identity gives the $A_{p}$ bound for the already given weight $w$ .

1.2F2givenalgebra

For $1 < p < q < \infty$ put $θ : = (p - 1) / (q - 1) \in (0, 1)$ and $σ : = w^{- 1 / (p - 1)}$ , so that $w^{- 1 / (q - 1)} = σ^{θ}$ . By the power-mean inequality of [F2], $⟨ σ^{θ} ⟩_{Q} \leq ⟨ σ ⟩_{Q}^{θ}$ for every cube $Q$ ; raising to the $(q - 1)$ -th power gives $⟨ w^{- 1 / (q - 1)} ⟩_{Q}^{q - 1} \leq ⟨ w^{- 1 / (p - 1)} ⟩_{Q}^{p - 1}$ .

1.3F1givenalgebra

For $w \in A_{1}$ and every cube $Q$ , put $m = {ess inf}_{Q} w$ . By [F1], $m \geq ⟨ w ⟩_{Q} / (c_{n} [w]_{A_{1}}) > 0$ . For every $q > 1$ , $w^{- 1 / (q - 1)} \leq m^{- 1 / (q - 1)}$ a.e. on $Q$ , so $⟨ w ⟩_{Q} ⟨ w^{- 1 / (q - 1)} ⟩_{Q}^{q - 1} \leq ⟨ w ⟩_{Q} / m \leq c_{n} [w]_{A_{1}}$ . Taking suprema proves $A_{1} \subseteq A_{q}$ with the stated bound for the entire range $q > 1$ .

2.1step 1.2givenalgebra

Combining step 1.2 with the definition, for every cube $Q$ one has $⟨ w ⟩_{Q} ⟨ w^{- 1 / (q - 1)} ⟩_{Q}^{q - 1} \leq ⟨ w ⟩_{Q} ⟨ w^{- 1 / (p - 1)} ⟩_{Q}^{p - 1} \leq [w]_{A_{p}}$ ; taking the supremum in $Q$ gives $[w]_{A_{q}} \leq [w]_{A_{p}}$ and in particular $A_{p} \subseteq A_{q}$ for $1 < p < q < \infty$ .

3.1step 1.1step 2.1step 1.3∎

Steps 2.1 and 1.3 are the two nesting assertions, and step 1.1 is the duality assertion together with the exact identity of characteristics; this proves the lemma with $C_{n} = c_{n}$ .

Depends on

Used by

The Aₚ classes are open in the exponent Corollary
The Muckenhoupt Aᵢnfinity class Definition
The Hardy-Littlewood maximal operator characterises Aₚ Theorem

Dependency tree · two levels

39 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)