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

Let X be an RD-space with μ(X ) = ∞, which means that X is a space of homogeneous type in the sense of Coifman and Weiss and its measure has the reverse doubling property. In this paper, we characterize the atomic Hardy spaces H at(X ) of Coifman and Weiss for p ∈ (n/(n + 1), 1] via the radial maximal function, where n is the “dimension” of X , and the range of index p is the best possible. This completely answers the question proposed by Ronald R. Coifman and Guido Weiss in 1977 in this setting, and improves on a deep result of Uchiyama in 1980 on an Ahlfors 1-regular space and a recent result of Loukas Grafakos et al in this setting. Moreover, we obtain a maximal function theory of localized Hardy spaces in the sense of Goldberg on RD-spaces by generalizing the above result to localized Hardy spaces and establishing the links between Hardy spaces and localized Hardy spaces. These results have a wide range of applications. In particular, we characterize the Hardy spaces H at(M) via the radial maximal function generated by the heat kernel of the Laplace-Beltrami operator ∆ on complete noncompact connected manifolds M having a doubling property and supporting a scaled Poincaré inequality for all p ∈ (n/(n + α), 1], where α represents the regularity of the heat kernel. This extends some recent results of Russ and Auscher-McIntosh-Russ. Mathematics Subject Classification (2000) Primary 42B30; Secondary 42B25, 42B35 Dachun Yang was supported by the National Natural Science Foundation (Grant No. 10871025) of China. D. Yang School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China e-mail: [email protected] Y. Zhou School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China e-mail: [email protected] Y. Zhou Department of Mathematics and Statistics, University of Jyväskylä, P. O. Box 35 (MaD), FI-40014, Finland e-mail: [email protected]