{
  "id": "0903.4725",
  "version": "v2",
  "published": "2009-03-27T01:59:36.000Z",
  "updated": "2009-06-08T12:45:49.000Z",
  "title": "Boundedness of Sublinear Operators on Product Hardy Spaces and Its Application",
  "authors": [
    "Der-Chen Chang",
    "Dachun Yang",
    "Yuan Zhou"
  ],
  "comment": "J. Math. Soc. Japan (to appear)",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $p\\in(0, 1]$. In this paper, the authors prove that a sublinear operator $T$ (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces $H^p({{\\mathbb R}^n\\times{\\mathbb R}^m})$ to some quasi-Banach space ${\\mathcal B}$ if and only if $T$ maps all $(p, 2, s_1, s_2)$-atoms into uniformly bounded elements of ${\\mathcal B}$. Here $s_1\\ge\\lfloor n(1/p-1)\\rfloor$ and $s_2\\ge\\lfloor m(1/p-1)\\rfloor$. As usual, $\\lfloor n(1/p-1)\\rfloor$ denotes the maximal integer no more than $n(1/p-1)$. Applying this result, the authors establish the boundedness of the commutators generated by Calder\\'on-Zygmund operators and Lipschitz functions from the Lebesgue space $L^p({{\\mathbb R}^n\\times{\\mathbb R}^m})$ with some $p>1$ or the Hardy space $H^p({{\\mathbb R}^n\\times{\\mathbb R}^m})$ with some $p\\le1$ but near 1 to the Lebesgue space $L^q({{\\mathbb R}^n\\times{\\mathbb R}^m})$ with some $q>1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-06-08T12:45:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B30",
      "42B25",
      "47B47"
    ],
    "keywords": [
      "product hardy spaces",
      "boundedness",
      "lebesgue space",
      "application",
      "quasi-banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.4725C"
    }
  }
}