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

7 pages