{
  "id": "1812.00817",
  "version": "v1",
  "published": "2018-11-30T17:38:29.000Z",
  "updated": "2018-11-30T17:38:29.000Z",
  "title": "Extension criterions for homogeneous Sobolev space of functions of one variable",
  "authors": [
    "Pavel Shvartsman"
  ],
  "comment": "53 pages. arXiv admin note: substantial text overlap with arXiv:1808.01467, arXiv:1710.07826",
  "categories": [
    "math.FA"
  ],
  "abstract": "For each $p>1$ and each positive integer $m$ we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset $E$ of the real line. We show that the classical one dimensional Whitney's extension operator is \"universal\" for the scale of $L^m_p(R)$ spaces in the following sense: for every $p\\in(1,\\infty]$ it provides almost optimal $L^m_p$-extensions of functions defined on $E$. The operator norm of this extension operator is bounded by a constant depending only on $m$. This enables us to prove several constructive $L^m_p$-extension criterions expressed in terms of $m^{th}$ order divided differences of functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-30T17:38:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35"
    ],
    "keywords": [
      "homogeneous sobolev space",
      "extension criterions",
      "dimensional whitneys extension operator",
      "intrinsic characterizations",
      "order divided differences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}