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