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

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