Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H _α ^p (X), H _d ^p (X), and H ^*,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide with L ^p (X) when p ∈ (1,∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H ^∗,p (X) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1),1], it is proved that the space H ^*,p (X), the Hardy space H ^p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H ^p (X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B. This work was supported by the National Science Foundation of USA (Grant No. DMS 0400387), the University of Missouri Research Council (Grant No. URC-07-067), the National Science Foundation for Distinguished Young Scholars of China (Grant No. 10425106) and the Program for New Century Excellent Talents in University of the Ministry of Education of China (Grant No. 04-0142)