Wavelet Characterization of Besov and Triebel--Lizorkin Spaces on Spaces of Homogeneous Type and Its Applications

TL;DR

This paper develops wavelet and molecular characterizations of Besov and Triebel--Lizorkin spaces on spaces of homogeneous type, removing reliance on reverse doubling and triangle inequality assumptions.

Contribution

It introduces wavelet and molecular characterizations of these function spaces on homogeneous type spaces without requiring reverse doubling or triangle inequality.

Findings

Wavelet characterization of Besov and Triebel--Lizorkin spaces established.

Boundedness of almost diagonal operators on sequence spaces proved.

Littlewood--Paley characterizations derived for Triebel--Lizorkin spaces.

Abstract

In this article, the authors establish the wavelet characterization of Besov and Triebel--Lizorkin spaces on a given space $(X,d,\mu)$ of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel--Lizorkin sequence spaces on $X$, and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of Besov and Triebel--Lizorkin spaces. Applying this molecular characterization, the authors further establish the Littlewood--Paley characterizations of Triebel--Lizorkin spaces on $X$. The main novelty of this article is that all these results get rid of their dependence on the reverse doubling property of $\mu$ and also the triangle inequality of $d$, by fully using the geometrical property of $X$ expressed via its equipped quasi-metric $d$, dyadic reference points, dyadic cubes, and wavelets