Some function spaces relative to Morrey-Campanato spaces on metric spaces

Abstract

In this paper, the author introduces the Morrey-Campanato spaces $L^{s}_{p}(X)$ and the spaces $C^{s}_{p}(X)$ on spaces of homogeneous type including metric spaces and some fractals, and establishes some embedding theorems between these spaces under some restrictions and the Besov spaces and the Triebel-Lizorkin spaces. In particular, the author proves that $L^{s}_{p}(X) = B^{s}_{\infty, \infty}(X)$ if $0<s<\infty$ and $\mu(X)<\infty$. The author also introduces some new function spaces $A^{s}_{p}(X)$ and $B^{s}_{p}(X)$ and proves that these new spaces when $0<s<1$ and $1<p<\infty$ are just the Triebel-Lizorkin space $F^{s}_{p, \infty}(X)$ if $X$ is a metric space, and the spaces $A^{1}_{p}(X)$ and $B^{1}_{p}(X)$ when $1<p\le\infty$ are just the Hajlasz-Sobolev spaces $W^{1}_{p}(X)$. Finally, as an application, the author gives a new characterization of the Hajlasz-Sobolev spaces by making use of the sharp maximal function.