Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Characterizations of Maximal Functions, Decompositions, and Dual Spaces

TL;DR

This paper develops a comprehensive theory of Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type, including characterizations, duality, and applications to variable Hardy spaces, overcoming geometric and measure-theoretic challenges.

Contribution

It introduces a new Hardy space framework on spaces of homogeneous type, establishing characterizations, dual spaces, and applications, even without reverse doubling or triangle inequality assumptions.

Findings

Established real-variable characterizations of $H_Y^*({ mf X})$

Identified the dual space as a ball Campanato-type space

Extended results to variable Hardy spaces with new techniques

Abstract

Let $({\mathcal X},\rho,\mu)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports a Fefferman--Stein vector-valued maximal inequality, and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_{Y}^*({\mathcal X})$, associated with $Y({\mathcal X})$, via the grand maximal function, and then establish its various real-variable characterizations, respectively, in terms of radial or non-tangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of $H_{Y}^*({\mathcal X})$, which proves to be a ball Campanato-type function space associated with $Y({\mathcal X})$. All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this article exist in that, to escape the reverse doubling condition of $\mu$ and the triangle inequality of $\rho$, the authors cleverly construct admissible sequences of balls, and fully use the geometrical properties of ${\mathcal X}$ expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of $H_{Y}^*({\mathcal X})$, the authors further prove that $Y({\mathcal X})$ can be embedded into the weighted Lebesgue space with certain special weight, and then can fully use the known results of the weighted Lebesgue space.