The maximal function on spaces of homogeneous type, or adjacent dyadic cubes do good

Abstract

We prove that the Hardy--Littlewood maximal operator $M$ is bounded on the variable Lebesgue space $L^{p(\cdot )}(X,d,\mu )$, with $1<p_{-}\le p_{+}<\infty$, over an unbounded space of homogeneous type $(X,d,\mu )$ with a Borel-semiregular measure $\mu$, if and only if the averaging operators $T_{\mathcal{Q}}$ are bounded on $L^{p(\cdot )}(X,d,\mu )$ uniformly over all families $\mathcal{Q}$ of pairwise disjoint ``cubes'' from a Hytönen--Kairema dyadic system on $X$. This extends Diening's well-known characterization of the boundedness of $M$ on $L^{p(\cdot )}({\mathbb{R}}^n)$ to the setting of spaces of homogeneous type, while also providing a slight refinement of the original result.