Sharp weighted bounds involving $A_{\infty}$

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

$${\left[w\right]}_{A_{\infty }}^{\prime }:=\underset{Q}{sup}\frac{1}{w\left(Q\right)}{\int }_{Q}M\left(w{\chi }_Q\right).$$

In particular, we show the following improvement of the first author’s $A_2$ theorem for Calderón–Zygmund operators $T$:

$$\parallel T{\parallel }_{\mathcal{ℬ}\left(L^2\left(w\right)\right)}\le c_T{\left[w\right]}_{A_2}^{1∕2}{\left({\left[w\right]}_{A_{\infty }}^{\prime }+{\left[w^{-1}\right]}_{A_{\infty }}^{\prime }\right)}^{1∕2}.$$

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

$$\begin{array}{llll}\hfill \parallel T{\parallel }_{\mathcal{ℬ}\left(L^p\left(w\right)\right)}& \le cp{p}^{\prime }{\left[w\right]}_{A_1}^{1∕p}{\left({\left[w\right]}_{A_{\infty }}^{\prime }\right)}^{1∕p^{\prime }},1<p<\infty ,& \hfill & \\ \hfill \parallel Tf{\parallel }_{L^{1,\infty }\left(w\right)}& \le c{\left[w\right]}_{A_1}log\left(e+{\left[w\right]}_{A_{\infty }}^{\prime }\right)\parallel f{\parallel }_{L^1\left(w\right)}.& \hfill & \end{array}$$

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