{
  "id": "0906.1316",
  "version": "v1",
  "published": "2009-06-07T00:59:00.000Z",
  "updated": "2009-06-07T00:59:00.000Z",
  "title": "Boundedness of Linear Operators via Atoms on Hardy Spaces with Non-doubling Measures",
  "authors": [
    "Dachun Yang",
    "Dongyong Yang"
  ],
  "comment": "Georgian Math. J. (to appear)",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "Let $\\mu$ be a non-negative Radon measure on ${\\mathbb R}^d$ which only satisfies the polynomial growth condition. Let ${\\mathcal Y}$ be a Banach space and $H^1(\\mu)$ the Hardy space of Tolsa. In this paper, the authors prove that a linear operator $T$ is bounded from $H^1(\\mu)$ to ${\\mathcal Y}$ if and only if $T$ maps all $(p, \\gamma)$-atomic blocks into uniformly bounded elements of ${\\mathcal Y}$; moreover, the authors prove that for a sublinear operator $T$ bounded from $L^1(\\mu)$ to $L^{1, \\infty}(\\mu)$, if $T$ maps all $(p, \\gamma)$-atomic blocks with $p\\in(1, \\infty)$ and $\\gamma\\in{\\mathbb N}$ into uniformly bounded elements of $L^1(\\mu)$, then $T$ extends to a bounded sublinear operator from $H^1(\\mu)$ to $L^1(\\mu)$. For the localized atomic Hardy space $h^1(\\mu)$, corresponding results are also presented. Finally, these results are applied to Calder\\'on-Zygmund operators, Riesz potentials and multilinear commutators generated by Calder\\'on-Zygmund operators or fractional integral operators with Lipschitz functions, to simplify the existing proofs in the corresponding papers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-07T00:59:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B30",
      "42B35"
    ],
    "keywords": [
      "non-doubling measures",
      "uniformly bounded elements",
      "atomic blocks",
      "sublinear operator",
      "calderon-zygmund operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.1316Y"
    }
  }
}