Weak type (1,1) bound criterion for singular integral with rough kernel and its applications

TL;DR

This paper establishes a weak type (1,1) bound criterion for singular integral operators with rough kernels and demonstrates its application to various important operators in harmonic analysis.

Contribution

It introduces a new weak type (1,1) bound criterion for rough kernel singular integrals and applies it to prove weak type (1,1) for several key operators.

Findings

Proved weak type (1,1) for Calderón commutator and its variants

Established the criterion's applicability to Muckenhoupt type singular integrals

Enhanced understanding of rough kernel operator bounds in harmonic analysis

Abstract

In this paper, a weak type (1,1) bound criterion is established for singular integral operator with rough kernel. As some applications of this criterion, we prove some important operators with rough kernel in harmonic analysis, such as Calder\'on commutator, higher order Calder\'on commutator, general Calder\'on commutator, Calder\'on commutator of Bajsanski-Coifman type and general singular integral of Muckenhoupt type, are all of weak type (1,1).