Sharp weighted bounds involving A∞

Hytönen, Tuomas; Pérez, Carlos

doi:10.2140/apde.2013.6.777

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Sharp weighted bounds involving $A_{\infty}$

Tuomas Hytönen and Carlos Pérez

Vol. 6 (2013), No. 4, 777–818

DOI: 10.2140/apde.2013.6.777

Abstract

We improve on several weighted inequalities of recent interest by replacing a part of the $A_{p}$ bounds by weaker $A_{\infty}$ estimates involving Wilson’s $A_{\infty}$ constant

{[w]}_{A_{\infty}}^{'} : = sup_{Q} \frac{1}{w (Q)} \int_{Q} M (w χ_{Q}) .

In particular, we show the following improvement of the first author’s $A_{2}$ theorem for Calderón–Zygmund operators $T$ :

∥ T ∥_{ℬ (L^{2} (w))} \leq c_{T} {[w]}_{A_{2}}^{1 ∕ 2} {({[w]}_{A_{\infty}}^{'} + {[w^{- 1}]}_{A_{\infty}}^{'})}^{1 ∕ 2} .

Corresponding $A_{p}$ type results are obtained from a new extrapolation theorem with appropriate mixed $A_{p}$ - $A_{\infty}$ bounds. This uses new two-weight estimates for the maximal function, which improve on Buckley’s classical bound.

We also derive mixed $A_{1}$ - $A_{\infty}$ type results of Lerner, Ombrosi and Pérez (2009) of the form

\begin{array}{l} ∥ T ∥_{ℬ (L^{p} (w))} & \leq c p p^{'} {[w]}_{A_{1}}^{1 ∕ p} {({[w]}_{A_{\infty}}^{'})}^{1 ∕ p^{'}}, 1 < p < \infty, \\ ∥ T f ∥_{L^{1, \infty} (w)} & \leq c {[w]}_{A_{1}} log (e + {[w]}_{A_{\infty}}^{'}) ∥ f ∥_{L^{1} (w)} . \end{array}

An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals.

Keywords

weighted norm inequalities, $A_p$ weights, sharp estimates, maximal function, Calderón–Zygmund operators

Mathematical Subject Classification 2010

Primary: 42B25

Secondary: 42B20, 42B35

Milestones

Received: 29 July 2011

Revised: 18 November 2011

Accepted: 19 November 2011

Published: 21 August 2013

Authors

Tuomas Hytönen
Department of Mathematics and Statistics University of Helsinki P.O. Box 68 FI-00014 Helsinki Finland

Carlos Pérez
Departamento de Análisis Matemático Facultad de Matemáticas Universidad De Sevilla 41080 Sevilla Spain

Vol. 6, No. 4, 2013