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

Article

DOI: 10.1007/s11425-008-0057-4

Cite this article as:
Grafakos, L., Liu, L. & Yang, D. Sci. China Ser. A-Math. (2008) 51: 2253. doi:10.1007/s11425-008-0057-4

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. For α ∈ (0, ∞) denote by Hαp (X), Hdp(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 Lp(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 Hp(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 Hp(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.

Keywords

space of homogeneous typeCalderón reproducing formulaspace of test functionmaximal functionHardy spaceatomLittlewood-Paley functionsublinear operatorquasi-Banach space

MSC(2000)

42B2542B3047B3847A30

Copyright information

© Science in China Press and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.School of Mathematical SciencesBeijing Normal UniversityBeijingChina
  3. 3.Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijingChina