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

Abstract

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 ϕ at (X ) and then establish its various maximal function characterizations.The authors also investigate the Littlewood-Paley characterizations of H ϕ 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 ϕ at (X ) and its improved version in case q < ∞, and their applications to criteria of the boundedness of sublinear operators from H ϕ 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 ϕ at (X ), namely, the Musielak-Orlicz BMO space BMO ϕ (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.