A Representation Theorem for Singular Integral Operators on Spaces of Homogeneous Type

TL;DR

This paper establishes a representation theorem for singular integral operators on spaces of homogeneous type, enabling a T(1) theorem in this setting by expressing operators as series of shifts, rearrangements, and paraproducts.

Contribution

It introduces a novel representation theorem for singular integrals on homogeneous spaces, extending T(1) theory to this context.

Findings

Representation of singular integrals as series of shifts and paraproducts

Proof of a T(1) theorem in spaces of homogeneous type

Extension of harmonic analysis tools to new geometric settings

Abstract

Let (X,d,\mu) be a space of homogeneous type and E a UMD Banach space. Under the assumption mu({x})=0 for all x in X, we prove a representation theorem for singular integral operators on (X,d,mu) as a series of simple shifts and rearrangements plus two paraproducts. This gives a T(1) Theorem in this setting.