# Proof of an extension of E. Sawyer's conjecture about weighted mixed weak-type estimates

## TL;DR

This paper proves a conjecture about weighted weak-type estimates involving Calderón-Zygmund operators and extends several related quantitative bounds, advancing understanding in harmonic analysis.

## Contribution

It confirms a conjecture on weighted weak-type estimates for Calderón-Zygmund operators with specific weight classes, extending prior results.

## Key findings

- Established the conjecture for $A_
abla$ weights.
- Extended quantitative estimates for weighted weak-type bounds.
- Applied to Hardy-Littlewood maximal function and Calderón-Zygmund operators.

## Abstract

We show that if $v\in A_\infty$ and $u\in A_1$, then there is a constant $c$ depending on the $A_1$ constant of $u$ and the $A_{\infty}$ constant of $v$ such that $$\Big\|\frac{ T(fv)} {v}\Big\|_{L^{1,\infty}(uv)}\le c\, \|f\|_{L^1(uv)},$$ where $T$ can be the Hardy-Littlewood maximal function or any Calder\'on-Zygmund operator. This result was conjectured in [IMRN, (30)2005, 1849--1871] and constitutes the most singular case of some extensions of several problems proposed by E. Sawyer and Muckenhoupt and Wheeden. We also improve and extends several quantitative estimates.