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 $$L^{p(\cdot )}(X,d,\mu )$$ L p ( · ) ( X , d , μ ) , with $$1 1 < p - ≤ p + < ∞ , over an unbounded space of homogeneous type $$(X,d,\mu )$$ ( X , d , μ ) with a Borel-semiregular measure $$\mu $$ μ , if and only if the averaging operators $$T_\mathcal {Q}$$ T Q are bounded on $$L^{p(\cdot )}(X,d,\mu )$$ L p ( · ) ( X , d , μ ) uniformly over all families $$\mathcal {Q}$$ Q of pairwise disjoint “cubes” from a Hytönen–Kairema dyadic system on X . This extends Diening’s well-known characterization of the boundedness of M on $$L^{p(\cdot )}(\mathbb {R}^n)$$ L p ( · ) ( R n ) to the setting of spaces of homogeneous type, while also providing a slight refinement of the original result.