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We show that the Hardy-Littlewood maximal operator is bounded on a reflexivevariable 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 correspondingresult of Lars Diening from the Euclidean setting of $\mathbb{R}^n$ to thesetting of spaces of homogeneous type $(X,d,\mu)$.

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