{
  "id": "1509.05820",
  "version": "v1",
  "published": "2015-09-18T23:41:36.000Z",
  "updated": "2015-09-18T23:41:36.000Z",
  "title": "Boundedness of Commutators on Hardy Spaces over Metric Measure Spaces of Non-homogeneous Type",
  "authors": [
    "Haibo Lin",
    "Suqing Wu",
    "Dachun Yang"
  ],
  "comment": "34 pages, Submitted",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $(\\mathcal{X},d,\\mu)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T$ be a Calder\\'{o}n-Zygmund operator with kernel satisfying only the size condition and some H\\\"ormander-type condition, and $b\\in\\rm{\\widetilde{RBMO}(\\mu)}$ (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $T_b:=bT-Tb$ generated by $T$ and $b$ from the atomic Hardy space $\\widetilde{H}^1(\\mu)$ with the discrete coefficient into the weak Lebesgue space $L^{1,\\,\\infty}(\\mu)$. The boundedness of the commutator generated by the generalized fractional integral $T_\\alpha\\,(\\alpha\\in(0,1))$ and the $\\rm{\\widetilde{RBMO}(\\mu)}$ function from $\\widetilde{H}^1(\\mu)$ into $L^{1/{(1-\\alpha)},\\,\\fz}(\\mu)$ is also presented. Moreover, by an interpolation theorem for sublinear operators, the authors show that the commutator $T_b$ is bounded on $L^p(\\mu)$ for all $p\\in(1,\\infty)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-18T23:41:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B47",
      "42B20",
      "42B35",
      "30L99"
    ],
    "keywords": [
      "metric measure space",
      "non-homogeneous type",
      "commutator",
      "boundedness",
      "discrete coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}