{
  "id": "1511.07598",
  "version": "v1",
  "published": "2015-11-24T07:31:13.000Z",
  "updated": "2015-11-24T07:31:13.000Z",
  "title": "Littlewood-Paley Characterizations of Fractional Sobolev Spaces via Averages on Balls",
  "authors": [
    "Feng Dai",
    "Jun Liu",
    "Dachun Yang",
    "Wen Yuan"
  ],
  "comment": "29 pages, Submitted",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "In this paper, the authors characterize Sobolev spaces $W^{\\alpha,p}({\\mathbb R}^n)$ with the smoothness order $\\alpha\\in(0,2]$ and $p\\in(\\max\\{1, \\frac{2n}{2\\alpha+n}\\},\\infty)$, via the Lusin area function and the Littlewood-Paley $g_\\lambda^\\ast$-function in terms of centered ball averages. The authors also show that the condition $p\\in(\\max\\{1, \\frac{2n}{2\\alpha+n}\\},\\infty)$ is nearly sharp in the sense that these characterizations are no longer true when $p\\in (1,\\max\\{1, \\frac{2n}{2\\alpha+n}\\})$. These characterizations provide a new possible way to introduce fractional Sobolev spaces with smoothness order in $(1,2]$ on metric measure spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-24T07:31:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "42B25",
      "42B20",
      "42B35"
    ],
    "keywords": [
      "fractional sobolev spaces",
      "littlewood-paley characterizations",
      "smoothness order",
      "metric measure spaces",
      "authors characterize sobolev spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}