Cancellative sparse domination

TL;DR

This paper introduces a general sparse domination principle that accounts for cancellative structures, providing new weighted bounds for martingales and Calderón-Zygmund operators across various measure spaces.

Contribution

It develops a unified sparse domination framework respecting cancellative properties, extending to measure spaces, martingale settings, and Euclidean spaces with quantitative weighted results.

Findings

Sparse domination results in general measure spaces.

A sparse characterization of the $H^1$ norm in martingale settings.

Quantitatively sharp weighted bounds for martingales and Calderón-Zygmund operators.

Abstract

We present a general sparse domination principle which respects the cancellative structure of the functions under study. We obtain sparse domination results in general measure spaces, including general martingale settings in one and two parameters, and in the Euclidean setting. In the one-parameter martingale setting, we obtain a sparse characterization of the $H^1$ norm. The proofs make critical use of precise level-set estimates for generalized versions of medians. Our results imply new, quantitatively sharp, weighted results for martingales and Calder\'on-Zygmund operators acting on $H^p$ spaces.