Maximal singular integral operators acting on noncommutative $L_p$-spaces

TL;DR

This paper develops boundedness criteria for noncommutative maximal Calderón-Zygmund operators, establishing new weak and strong type inequalities, including for rough kernels, advancing the understanding of noncommutative harmonic analysis.

Contribution

It introduces the first noncommutative maximal inequalities for families of linear operators not reducible to positive ones and extends strong type estimates to rough kernels.

Findings

Established weak type (1,1) estimates for noncommutative maximal operators.

Proved strong type (p,p) estimates for rough kernels with 1<p<∞.

Provided affirmative results for noncommutative Calderón-Zygmund operators under integral regularity conditions.

Abstract

In this paper, we study the boundedness theory for maximal Calder\'on-Zygmund operators acting on noncommutative $L_p$-spaces. Our first result is a criterion for the weak type $(1,1)$ estimate of noncommutative maximal Calder\'on-Zygmund operators; as an application, we obtain the weak type $(1,1)$ estimates of operator-valued maximal singular integrals of convolution type under proper {regularity} conditions. These are the {\it first} noncommutative maximal inequalities for families of linear operators that can not be reduced to positive ones. For homogeneous singular integrals, the strong type $(p,p)$ ($1<p<\infty$) maximal estimates are shown to be true even for {rough} kernels. As a byproduct of the criterion, we obtain the noncommutative weak type $(1,1)$ estimate for Calder\'on-Zygmund operators with integral regularity condition that is slightly stronger than the H\"ormander condition; this evidences somewhat an affirmative answer to an open question in the noncommutative Calder\'on-Zygmund theory.