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

arXiv commentManuscripta Math., to appear

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