Noncommutative maximal operators with rough kernels

TL;DR

This paper establishes weak type (1,1) bounds for noncommutative maximal operators with rough kernels using noncommutative Calderón-Zygmund decomposition and microlocal analysis.

Contribution

It introduces a novel approach to handle rough kernels in noncommutative maximal operators, proving weak type (1,1) boundedness.

Findings

Proved weak type (1,1) boundedness for noncommutative maximal operators with rough kernels.

Developed a microlocal decomposition technique for rough kernel analysis.

Applied noncommutative Calderón-Zygmund decomposition in this context.

Abstract

This paper is devoted to the study of noncommutative maximal operators with rough kernels. More precisely, we prove the weak type $(1,1)$ boundedness for noncommutative maximal operators with rough kernels. The proof of weak type (1,1) estimate is based on the noncommutative Calder\'on-Zygmund decomposition. To deal with the rough kernel, we use the microlocal decomposition in the proofs of both the bad and good functions.