Pointwise Characterization of Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type

TL;DR

This paper provides a new pointwise characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type, removing previous measure and metric restrictions, and extends results to inhomogeneous spaces even in RD-spaces.

Contribution

It introduces a novel pointwise characterization of these function spaces that does not rely on reverse doubling or metric conditions, broadening applicability.

Findings

Established pointwise characterizations without reverse doubling condition.

Extended characterization to inhomogeneous spaces in RD-spaces.

Clarified relationships among various Hajlasz and classical function spaces.

Abstract

In this article, the authors establish the pointwise characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type via clarifying the relationship among Haj\l asz-Sobolev spaces, Haj\l asz-Besov and Haj\l asz-Triebel-Lizorkin spaces, grand Besov and Triebel-Lizorkin spaces, and Besov and Triebel-Lizorkin spaces. A major novelty of this article is that all results presented in this article get rid of both the dependence on the reverse doubling condition of the measure and the metric condition of the quasi-metric under consideration. Moreover, the pointwise characterization of the inhomogeneous version is new even when the underlying space is an RD-space.