{
  "id": "1607.01660",
  "version": "v1",
  "published": "2016-07-06T15:09:45.000Z",
  "updated": "2016-07-06T15:09:45.000Z",
  "title": "Whitney-type extension theorems for jets generated by Sobolev functions. I",
  "authors": [
    "Pavel Shvartsman"
  ],
  "comment": "75 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $L^m_p(R^n)$, $p\\in [1,\\infty]$, be the homogeneous Sobolev space, and let $E\\subset R^n$ be a closed set. For each $p>n$ and each non-negative integer $m$ we give an intrinsic characterization of the restrictions to $E$ of $m$-jets generated by functions $F\\in L^{m+1}_p(R^n)$. Our trace criterion is expressed in terms of variations of corresponding Taylor remainders of $m$-jets evaluated on a certain family of \"well separated\" two point subsets of $E$. For $p=\\infty$ this result coincides with the classical Whitney-Glaeser extension theorem for $m$-jets. Our approach is based on a representation of the Sobolev space $L^{m+1}_p(R^n)$, $p>n$, as a union of $C^{m,(d)}(R^n)$-spaces where $d$ belongs to a family of metrics on $R^n$ with certain \"nice\" properties. Here $C^{m,(d)}(R^n)$ is the space of $C^m$-functions on $R^n$ whose partial derivatives of order $m$ are Lipschitz functions with respect to $d$. This enables us to show that, for every non-negative integer $m$ and every $p\\in (n,\\infty)$, the very same classical linear Whitney extension operator provides an almost optimal extension of $m$-jets generated by $L^{m+1}_p$-functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-06T15:09:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35"
    ],
    "keywords": [
      "whitney-type extension theorems",
      "sobolev functions",
      "classical linear whitney extension operator",
      "classical whitney-glaeser extension theorem",
      "non-negative integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 75,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}