Weighted weak $(1,1)$ estimate for non-commutative square function

TL;DR

This paper proves a weighted weak (1,1) inequality for a non-commutative square function related to ball averages and martingales, extending previous results to a weighted context.

Contribution

It introduces a weighted weak (1,1) inequality for non-commutative square functions, advancing the understanding of weighted inequalities in non-commutative harmonic analysis.

Findings

Establishes a weighted weak (1,1) inequality for non-commutative square functions.

Extends previous unweighted results to the weighted setting.

Provides new tools for analysis in non-commutative harmonic analysis.

Abstract

In this article, we consider weighted weak type $(1,1)$ inequality for certain square function associated to differences of ball averages and martingale in the non-commutative setting. This establishes a weighted version of main result of \cite{hong2021noncommutative}.