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 the Hyt\"onen--Kairema dyadic system with a distinguished center point. This is an analogue, in the setting of spaces of homogeneous type, of Diening's well-known characterization of the boundedness of $M$ on $L^{p(\cdot)}(\mathbb{R}^n)$.