  Given a space of homogeneous type we give sufficient conditions on a variable
exponent {p(.)} so that the fractional maximal operator {M_{\eta}} maps
{L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = {\eta}}. In the
endpoint case we also prove the corresponding weak type inequality. As an
application we prove norm inequalities for the fractional integral operator
{I_{\eta}}. Our proof for the fractional maximal operator uses the theory of
dyadic cubes on spaces of homogeneous type, and even in the Euclidean setting
it is simpler than existing proofs. For the fractional integral operator we
extend a pointwise inequality of Welland to spaces of homogeneous type. Our
work generalizes results from the Euclidean case and extends recent work by
Adamowicz, et al. on the Hardy-Littlewood maximal operator on spaces of
homogeneous type.
