Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type with Applications to Boundedness of Calder\'on-Zygmund Operators

TL;DR

This paper develops Besov and Triebel-Lizorkin spaces on spaces of homogeneous type, proving their independence from certain choices and establishing boundedness of Calderón-Zygmund operators without relying on reverse doubling conditions.

Contribution

It introduces a new real-variable theory of these function spaces on spaces of homogeneous type, independent of reverse doubling assumptions, and applies it to operator boundedness.

Findings

Spaces are independent of exp-ATI choices

Boundedness of Calderón-Zygmund operators established

Coincidence with known function spaces under certain conditions

Abstract

In this article, the authors introduce Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss, prove that these (in)homogeneous Besov and Triebel-Lizorkin spaces are independent of the choices of both exp-ATIs (or exp-IATIs) and underlying spaces of distributions, and give some basic properties of these spaces. As applications, the authors show that some known function spaces coincide with certain special cases of Besov and Triebel-Lizorkin spaces and, moreover, obtain the boundedness of Calder\'on-Zygmund operators on these Besov and Triebel-Lizorkin spaces. All these results strongly depend on the geometrical properties, reflected via its dyadic cubes, of the considered space of homogeneous type. Comparing with the known theory of these spaces on metric measure spaces, a major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying space and hence give a final real-variable theory of these function spaces on spaces of homogeneous type.