Two weight commutators on spaces of homogeneous type and applications

TL;DR

This paper develops a comprehensive theory for two weight commutators of Calderón--Zygmund operators on spaces of homogeneous type, introducing sparse domination techniques and applying them to various classical operators.

Contribution

It establishes the boundedness of two weight commutators using sparse domination and Haar basis on spaces of homogeneous type, extending classical results to a broader setting.

Findings

Proved sparse operator domination of commutators.

Established two weight commutator theorems for multiple operators.

Extended classical results to spaces of homogeneous type.

Abstract

In this paper, we establish the two weight commutator of Calder\'on--Zygmund operators in the sense of Coifman--Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for $A_2$ weight and by proving the sparse operator domination of commutators. The main tool here is the Haar basis and the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutator) for the following Calder\'on--Zygmund operators: Cauchy integral operator on $\mathbb R$, Cauchy--Szeg\"o projection operator on Heisenberg groups, Szeg\"o projection operators on a family of unbounded weakly pseudoconvex domains, Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (one-dimension and high dimension).