{
  "id": "0904.4521",
  "version": "v2",
  "published": "2009-04-29T01:46:11.000Z",
  "updated": "2009-08-30T03:23:13.000Z",
  "title": "Radial Maximal Function Characterizations of Hardy Spaces on RD-Spaces and Their Applications",
  "authors": [
    "Dachun Yang",
    "Yuan Zhou"
  ],
  "comment": "Math. Ann., to appear",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "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 $\\Delta$ on complete noncompact connected manifolds $M$ having a doubling property and supporting a scaled Poincar\\'e inequality for all $p\\in(n/(n+\\alpha),1]$, where $\\alpha$ represents the regularity of the heat kernel. This extends some recent results of Russ and Auscher-McIntosh-Russ.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-08-30T03:23:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B30",
      "42B25",
      "42B35"
    ],
    "keywords": [
      "radial maximal function characterizations",
      "localized hardy spaces",
      "applications",
      "heat kernel",
      "doubling property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.4521Y"
    }
  }
}