Quantitative weighted mixed weak-type inequalities for classical operators

TL;DR

This paper advances the understanding of weighted weak-type inequalities for classical operators, providing more precise quantitative estimates involving weight constants, improving upon previous results by Muckenhoupt, Wheeden, and Sawyer.

Contribution

It introduces new quantitative bounds for mixed weak-type inequalities for Hardy-Littlewood maximal and Calderón-Zygmund operators, emphasizing the role of $A_p$ and $A_$ constants.

Findings

Enhanced inequalities with explicit weight constant dependence

More precise estimates for maximal and Calderón-Zygmund operators

Extension of previous results by Muckenhoupt, Wheeden, and Sawyer

Abstract

We improve on several mixed weak type inequalities both for the Hardy-Littlewood maximal function and for Calder\'on-Zygmund operators. These type of inequalities were considered by Muckenhoupt and Wheeden and later on by Sawyer estimating the $L^{1, \infty}(uv)$ norm of $v^{-1}T(fv)$ for special cases. The emphasis is made in proving new and more precise quantitative estimates involving the $A_p$ or $A_\infty$ constants of the weights involved.