Noncommutative weak type $(1,1)$ estimates for Calder\'on-Zygmund operators with a class of $L_1$-integral conditions

TL;DR

This paper develops a new noncommutative Calderón-Zygmund decomposition and proves weak type (1,1) bounds for noncommutative Calderón-Zygmund operators under near-$L_1$-Dini conditions.

Contribution

It introduces a refined noncommutative Calderón-Zygmund decomposition and establishes weak type (1,1) estimates under new $L_1$-integral conditions.

Findings

Established weak type (1,1) boundedness for noncommutative Calderón-Zygmund operators.

Introduced a novel splitting technique for the bad function in the decomposition.

Extended the class of $L_1$-conditions under which boundedness holds.

Abstract

We construct a slightly new noncommutative Calder\'on-Zygmund decomposition by further splitting the bad function. Using this tool, we prove the weak type (1,1) boundedness of noncommutative Calder\'on-Zygmund operators under a class of $L_1$-integral conditions, which are close to $L_1$-Dini conditions.