REAL-VARIABLE CHARACTERIZATIONS OF MUSIELAK-ORLICZ HARDY SPACES ON SPACES OF HOMOGENEOUS TYPE

Let (chi, d, mu) 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 (chi, d, mu). To be precise, the authors first introduce the atomic Musielak-Orlicz Hardy space H-at(phi)(chi) and then establish its various maximal function characterizations. The authors also investigate the Littlewood-Paley characterizations of H-at(phi)(chi) via Lusin area functions, Littlewood- Paley g-functions and Littlewood-Paley g(lambda)*-functions. The authors further obtain the finite atomic characterization of H-at(phi)(chi) and its improved version in case q < infinity, and their applications to criteria of the boundedness of sublinear operators from H-at(phi)(chi) to a quasi-Banach space, which are also applied to the boundedness of Calderon-Zygmund operators. Moreover, the authors find the dual space of H-at(phi)(chi), namely, the Musielak-Orlicz BMO space BMW phi(chi), present its several equivalent characterizations, and apply it to establish a new characterization of the set of pointwise multipliers for the space BMO(chi). The main novelty of this article is that, throughout the article, except the last section, mu is not assumed to satisfy the reverse doubling condition.