  In this article, the authors introduce Besov and Triebel-Lizorkin spaces on
spaces of homogeneous type in the sense of Coifman and Weiss, prove that these
(in)homogeneous Besov and Triebel-Lizorkin spaces are independent of the
choices of both exp-ATIs (or exp-IATIs) and underlying spaces of distributions,
and give some basic properties of these spaces. As applications, the authors
show that some known function spaces coincide with certain special cases of
Besov and Triebel-Lizorkin spaces and, moreover, obtain the boundedness of
Calder\'on-Zygmund operators on these Besov and Triebel-Lizorkin spaces. All
these results strongly depend on the geometrical properties, reflected via its
dyadic cubes, of the considered space of homogeneous type. Comparing with the
known theory of these spaces on metric measure spaces, a major novelty of this
article is that all results presented in this article get rid of the dependence
on the reverse doubling assumption of the considered measure of the underlying
space and hence give a final real-variable theory of these function spaces on
spaces of homogeneous type.
