{
  "id": "1507.08004",
  "version": "v1",
  "published": "2015-07-29T01:38:46.000Z",
  "updated": "2015-07-29T01:38:46.000Z",
  "title": "Characterizations of Besov and Triebel-Lizorkin Spaces via Averages on Balls",
  "authors": [
    "Feng Dai",
    "Amiran Gogatishvili",
    "Dachun Yang",
    "Wen Yuan"
  ],
  "comment": "21 pages, Submitted",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $\\ell\\in\\mathbb{N}$ and $p\\in(1,\\infty]$. In this article, the authors prove that the sequence $\\{f-B_{\\ell,2^{-k}}f\\}_{k\\in\\mathbb{Z}}$ consisting of the differences between $f$ and the ball average $B_{\\ell,2^{-k}}f$ characterizes the Besov space $\\dot B^\\alpha_{p,q}(\\rn)$ with $q\\in (0, \\infty]$ and the Triebel-Lizorkin space $\\dot F^\\alpha_{p,q}(\\rn)$ with $q\\in (1,\\infty]$ when the smoothness order $\\alpha\\in(0,2\\ell)$. More precisely, it is proved that $f-B_{\\ell,2^{-k}}f$ plays the same role as the approximation to the identity $\\varphi_{2^{-k}}\\ast f$ appearing in the definitions of $\\dot B^\\alpha_{p,q}(\\rn)$ and $\\dot F^\\alpha_{p,q}(\\rn)$. The corresponding results for inhomogeneous Besov and Triebel-Lizorkin spaces are also obtained. These results, for the first time, give a way to introduce Besov and Triebel-Lizorkin spaces with any smoothness order in $(0, 2\\ell)$ on spaces of homogeneous type, where $\\ell\\in{\\mathbb N}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-29T01:38:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "42B25",
      "42B35"
    ],
    "keywords": [
      "triebel-lizorkin space",
      "smoothness order",
      "characterizations",
      "ball average",
      "besov space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}