Weighted norm estimates of noncommutative Calder\'{o}n-Zygmund operators

TL;DR

This paper establishes weighted endpoint estimates for noncommutative Calderón-Zygmund operators, including weak-type (1,1) bounds and H1-L1 inequalities, under Muckenhoupt A1 weights and relaxed kernel regularity conditions.

Contribution

It introduces new weighted endpoint estimates for noncommutative singular integrals with weaker kernel regularity assumptions.

Findings

Weighted weak-type (1,1) estimate for noncommutative maximal Calderón-Zygmund operators

Weighted H1-L1 type inequality for operator-valued singular integrals

Results hold under Muckenhoupt A1 weights and relaxed kernel regularity conditions

Abstract

This paper is devoted to studying weighted endpoint estimates of operator-valued singular integrals. Our main results include weighted weak-type $(1,1)$ estimate of noncommutative maximal Calder\'{o}n-Zygmund operators, corresponding version of square functions and a weighted $H_1- L_1$ type inequality. All these results are obtained under the condition that the weight belonging to the Muchenhoupt $A_1$ class and certain regularity assumptions imposed on kernels which are weaker than the Lipschitz condition.