The maximal function on spaces of homogeneous type, or adjacent dyadic cubes do good
Alina Shalukhina
TL;DR
This work extends Diening’s Euclidean criterion for the boundedness of the Hardy--Littlewood maximal operator from $\mathbb{R}^n$ to spaces of homogeneous type by using the Hytönen–Kairema dyadic system. The central result shows that $M$ is bounded on $L^{p(\cdot)}(X,d,μ)$, with $1<p_-\le p_+<\infty$, if and only if the dyadic averaging operators $T_\mathcal{Q}$ are uniformly bounded on $L^{p(\cdot)}(X,d,μ)$ for all disjoint cube families $\mathcal{Q}$ drawn from the HK system. The paper develops a Musielak–Orlicz framework with generalized $\Phi$-functions, introduces the self-improving class $\mathscr{A^D}$ and its stronger variant $\mathscr{A}^{\mathscr{D}}_{strong}$, and proves a sufficient condition for $M$’s boundedness via strong domination. For variable Lebesgue exponents, the authors prove that $\varphi_{p(\cdot)}\in\mathscr{A^D}$ iff $\varphi_{p(\cdot)}\in\mathscr{A}^{\mathscr{D}}_{strong}$, yielding a refined, dyadic-structure-driven criterion that also specializes to the Euclidean setting with a corollary improving Diening’s theorem. These results provide a robust foundation for extrapolation-type arguments and operator boundedness in variable-exponent spaces on spaces of homogeneous type.
Abstract
We prove that the Hardy--Littlewood maximal operator $M$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(X,d,μ)$, with $1<p_-\le p_+<\infty$, over an unbounded space of homogeneous type $(X,d,μ)$ with a Borel-semiregular measure $μ$, if and only if the averaging operators $T_\mathcal{Q}$ are bounded on $L^{p(\cdot)}(X,d,μ)$ 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.