Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications

Abstract

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 .F or α ∈ (0, ∞ )d enote byH p(X), H p d(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 Calderon reproducing formula, it is shown that all these Hardy spaces coincide with L p (X )w henp ∈ (1, ∞) and with each other when p ∈ (n/(n +1 ) ,1). An atomic characterization for H ∗ ,p (X )w ithp ∈ (n/(n +1 ),1) is also established; moreover, in the range p ∈ (n/(n +1 ), 1), it is proved that the spaceH ∗ ,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman and Weiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X )t o 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.