A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot )}$ over reverse doubling spaces of homogeneous type

Let $(X,d,\mu )$ be a space of homogeneous type and $p(\cdot ):X\to [1,\mathrm{\infty }]$ be a variable exponent. We show that if the measure $\mu$ is Borel-semiregular and reverse doubling, then the condition ${\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{f}}_{x\in X}p(x)>1$ is necessary for the boundedness of the Hardy-Littlewood maximal operator $M$ on the variable Lebesgue space $L^{p(\cdot )}(X,d,\mu )$.