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 Hyt\"onen--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.