#### Vol. 6, No. 4, 2013

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Sharp weighted bounds involving $A_{\infty}$

### Tuomas Hytönen and Carlos Pérez

Vol. 6 (2013), No. 4, 777–818
##### 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

 ${\left[w\right]}_{{A}_{\infty }}^{\prime }:=\underset{Q}{sup}\frac{1}{w\left(Q\right)}{\int }_{Q}\phantom{\rule{0.3em}{0ex}}M\left(w{\chi }_{Q}\right).$

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

 $\parallel T{\parallel }_{\mathsc{ℬ}\left({L}^{2}\left(w\right)\right)}\le {c}_{T}\phantom{\rule{0.3em}{0ex}}{\left[w\right]}_{{A}_{2}}^{1∕2}{\left({\left[w\right]}_{{A}_{\infty }}^{\prime }+{\left[{w}^{-1}\right]}_{{A}_{\infty }}^{\prime }\right)}^{1∕2}.$

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}{llll}\hfill \parallel T{\parallel }_{\mathsc{ℬ}\left({L}^{p}\left(w\right)\right)}& \le cp{p}^{\prime }{\left[w\right]}_{{A}_{1}}^{1∕p}{\left({\left[w\right]}_{{A}_{\infty }}^{\prime }\right)}^{1∕{p}^{\prime }},\phantom{\rule{1em}{0ex}}1

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