New properties of Besov and Triebel-Lizorkin spaces on RD-spaces

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.