{
  "id": "1603.01781",
  "version": "v1",
  "published": "2016-03-06T03:17:04.000Z",
  "updated": "2016-03-06T03:17:04.000Z",
  "title": "Variable Weak Hardy Spaces and Their Applications",
  "authors": [
    "Xianjie Yan",
    "Dachun Yang",
    "Wen Yuan",
    "Ciqiang Zhuo"
  ],
  "comment": "60 pages, submitted",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $p(\\cdot):\\ \\mathbb R^n\\to(0,\\infty)$ be a variable exponent function satisfying the globally log-H\\\"older continuous condition. In this article, the authors first introduce the variable weak Hardy space on $\\mathbb R^n$, $W\\!H^{p(\\cdot)}(\\mathbb R^n)$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $W\\!H^{p(\\cdot)}(\\mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_{\\lambda}^\\ast$-function. As an application, the authors establish the boundedness of convolutional $\\delta$-type and non-convolutional $\\gamma$-order Calder\\'on-Zygmund operators from $H^{p(\\cdot)}(\\mathbb R^n)$ to $W\\!H^{p(\\cdot)}(\\mathbb R^n)$ including the critical case $p_-={n}/{(n+\\delta)}$, where $p_-:=\\mathop\\mathrm{ess\\,inf}_{x\\in \\rn}p(x).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-06T03:17:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B30",
      "42B25",
      "42B20",
      "42B35",
      "46E30"
    ],
    "keywords": [
      "variable weak hardy space",
      "application",
      "radial grand maximal function",
      "non-tangential maximal function characterizations",
      "lusin area function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160301781Y"
    }
  }
}