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

Abstract

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