Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type

Let (X , d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors establish a complete real-variable theory of Musielak–Orlicz Hardy spaces on (X , d, μ). To be precise, the authors first introduce the atomic Musielak–Orlicz Hardy space H at (X ) and then establish its various maximal function characterizations. The authors also investigate the Littlewood–Paley characterizations of H at (X ) via Lusin area functions, Littlewood– Paley g-functions and Littlewood–Paley g∗ λ-functions. The authors further obtain the finite atomic characterization of H at (X ) and its improved version in case q < ∞, and their applications to criteria of the boundedness of sublinear operators from H at (X ) to a quasi-Banach space, which are also applied to the boundedness of Calderón–Zygmund operators. Moreover, the authors find the dual space of H at (X ), namely, the Musielak–Orlicz BMO space BMO(X ), present its several equivalent characterizations, and apply it to establish a new characterization of the set of pointwise multipliers for the space BMO(X ). The main novelty of this article is that, throughout the article, except the last section, μ is not assumed to satisfy the reverse doubling condition.