{
  "id": "1904.03946",
  "version": "v1",
  "published": "2019-04-08T10:59:03.000Z",
  "updated": "2019-04-08T10:59:03.000Z",
  "title": "Metric characterization of the sum of fractional Sobolev spaces",
  "authors": [
    "Rémy Rodiac",
    "Jean Van Schaftingen"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We introduce a non-linear criterion which allows us to determine when a function can be written as a sum of functions belonging to homogeneous fractional spaces: for $\\ell \\in \\mathbb{N}^*$, $s_i\\in (0, 1)$ and $p_i \\in [1, +\\infty)$, $u : \\Omega \\to \\mathbb{R}$ can be decomposed as $u = u_1+\\dotsc+u_\\ell$ with $u_i \\in \\dot{W}^{s_i,p_i}(\\Omega)$ if and only if $$ \\iint\\limits_{\\Omega \\times \\Omega} \\min_{1 \\le i \\le \\ell} \\frac{|u (x) - u (y)|^{p_i}}{|x - y|^{n+s_ip_i}}\\,\\mathrm{d}x \\,\\mathrm{d}y <+\\infty. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-08T10:59:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35"
    ],
    "keywords": [
      "fractional sobolev spaces",
      "metric characterization",
      "non-linear criterion",
      "homogeneous fractional spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}