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

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Abstract

Let \({{\mathcal X}}\) be an RD-space with \({\mu({\mathcal X})=\infty}\), which means that \({{\mathcal X}}\) 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 \({H^p_{\rm at}({\mathcal X})}\) of Coifman and Weiss for \({p\in(n/(n+1),1]}\) via the radial maximal function, where n is the “dimension” of \({{\mathcal X}}\), 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 \({H^p_{\rm at}(M)}\) via the radial maximal function generated by the heat kernel of the Laplace-Beltrami operator Δ on complete noncompact connected manifolds M having a doubling property and supporting a scaled Poincaré inequality for all \({p\in(n/(n+\alpha),1]}\), where α represents the regularity of the heat kernel. This extends some recent results of Russ and Auscher-McIntosh-Russ.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, 100875, Beijing, People’s Republic of China

    Dachun Yang & Yuan Zhou

  2. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014, Jyväskylä, Finland

    Yuan Zhou

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  1. Dachun Yang
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  2. Yuan Zhou
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Correspondence to Dachun Yang.

Additional information

Dachun Yang was supported by the National Natural Science Foundation (Grant No. 10871025) of China.

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Yang, D., Zhou, Y. Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications. Math. Ann. 346, 307–333 (2010). https://doi.org/10.1007/s00208-009-0400-2

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  • DOI: https://doi.org/10.1007/s00208-009-0400-2

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