  We consider a version of M. Riesz fractional integral operator on a space of
homogeneous type and show an analogue of the well-known
Hardy--Littlewood--Sobolev theorem in this context. In our main result, we
investigate the dependence of the operator norm on weighted spaces on the
weight constant, and find the relationship between these two quantities. It it
shown that the estimate obtained is sharp in any given space of homogeneous
type with infinitely many points. Our result generalizes the recent Euclidean
result by Lacey, Moen, P{\'e}rez and Torres.
