In this paper, the author introduces the Morrey-Campanato spaces Lsp(X) and the spaces Cps(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 Lsp(X) = Bs∞,∞(X) if 0 < s < ∞ and µ(X) < ∞. The author also introduces some new function spaces Asp(X) and Bsp(X) and proves that these new spaces when 0 < s < 1 and 1 < p < ∞ are just the Triebel-Lizorkin space Fsp,∞(X) if X is a metric space, and the spaces A1p(X) and B1p(X) when 1 < p < ∞ are just the Hajłasz-Sobolev spaces W1p(X). Finally, as an application, the author gives a new characterization of the Hajłasz-Sobolev spaces by making use of the sharp maximal function.
