Dyadic representation and boundedness of non-homogeneous Calder\'on--Zygmund operators with mild kernel regularity

TL;DR

This paper introduces a new dyadic representation theorem that enables the proof of non-homogeneous T(1) theorems under weaker kernel regularity conditions, advancing the understanding of Calderón--Zygmund operators.

Contribution

It presents a novel dyadic representation theorem that relaxes kernel regularity assumptions for non-homogeneous Calderón--Zygmund operators.

Findings

Established a new dyadic representation theorem.

Proved a non-homogeneous T(1) theorem with weaker kernel regularity.

Extended the applicability of Calderón--Zygmund theory.

Abstract

We prove a new dyadic representation theorem with applications to the $T(1)$ and $A_2$ theorems. In particular, we obtain the non-homogeneous $T(1)$ theorem under weaker kernel regularity than the earlier approaches.