# A sparse approach to mixed weak type inequalities

## TL;DR

This paper introduces a sparse domination method to obtain quantitative mixed-type estimates for Calderón-Zygmund operators and related singular integrals, extending endpoint estimate techniques.

## Contribution

It provides a novel sparse domination framework for mixed weak type inequalities, advancing the approach to endpoint estimates in harmonic analysis.

## Key findings

- Established new mixed weak type inequalities under $uv\in A_{\infty}$ conditions.
- Extended sparse domination techniques to endpoint estimates.
- Enhanced understanding of the behavior of singular integrals with mixed weights.

## Abstract

In this paper we provide some quantitative mixed-type estimates assuming conditions that imply that $uv\in A_{\infty}$ for Calder\'on-Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced by Domingo-Salazar, Lacey and Rey and extended in works by Lerner, Ombrosi and the second author and Li, Perez, the second author and Roncal.