Hardy-Littlewood maximal operator on reflexive variable Lebesgue spaces over spaces of homogeneous type

We show that the Hardy-Littlewood maximal operator is bounded on a reflexive variable Lebesgue space L-p(.) over a space of homogeneous type (X, d, mu) if and only if it is bounded on its dual space L-p'(.), where 1/p(x) + 1/p' (x) = 1 for x is an element of X. This result extends the corresponding result of Lars Diening from the Euclidean setting of R-n to the setting of spaces (X, d, mu) of homogeneous type.