New Properties of Besov and Triebel-Lizorkin Spaces on RD-Spaces

TL;DR

This paper explores new properties and characterizations of Besov and Triebel-Lizorkin spaces on RD-spaces, including independence of definitions from regularity parameters and norm characterizations involving local Hardy spaces.

Contribution

It provides several equivalent characterizations of RD-spaces and demonstrates the independence of test function and space definitions from regularity choices, advancing the understanding of these function spaces.

Findings

Equivalent characterizations of RD-spaces.

Independence of space definitions from regularity parameter.

Norm characterizations involving local Hardy spaces.

Abstract

An RD-space $\mathcal X$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal X$. In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on $\mathcal X$ are independent of the choice of the regularity $\epsilon\in (0,1)$; as a result of this, the Besov and Triebel-Lizorkin spaces on $\mathcal X$ are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogeneous Besov and Triebel-Lizorkin spaces by the norms of homogeneous Besov and Triebel-Lizorkin spaces together with the norm of local Hardy spaces in the sense of Goldberg. Also, the authors obtain the sharp locally integrability of elements in Besov and Triebel-Lizorkin spaces.