Endpoint weak-type bounds beyond Calder\'on-Zygmund theory

TL;DR

This paper establishes weighted weak-type bounds for operators with limited-range sparse domination, improving understanding of Calderón-Zygmund operators and providing a new proof for sharp bounds involving weights.

Contribution

It introduces new weighted weak-type estimates for operators with limited-range sparse domination, extending beyond classical Calderón-Zygmund theory.

Findings

Weighted weak-type (r,r) estimates for limited-range sparse operators

Improved bounds for operators with square function sparse domination

A new simple proof of sharp bounds for Calderón-Zygmund operators

Abstract

We prove weighted weak-type $(r,r)$ estimates for operators satisfying $(r,s)$ limited-range sparse domination of $\ell^q$-type. Our results contain improvements for operators satisfying limited-range and square function sparse domination. In the case of operators $T$ satisfying standard sparse form domination such as Calder\'on-Zygmund operators, we provide a new and simple proof of the sharp bound $$ \|T\|_{L^1_w(\mathbf{R}^d)\rightarrow L^{1,\infty}_w(\mathbf{R}^d)} \lesssim [w]_1(1+\log [w]_{\text{FW}}). $$