{
  "id": "1608.00857",
  "version": "v1",
  "published": "2016-08-02T15:02:03.000Z",
  "updated": "2016-08-02T15:02:03.000Z",
  "title": "Sobolev extensions of Lipschitz mappings into metric spaces",
  "authors": [
    "Scott Zimmerman"
  ],
  "comment": "18 pages",
  "categories": [
    "math.MG"
  ],
  "abstract": "Wenger and Young proved that the pair $(\\mathbb{R}^m,\\mathbb{H}^n)$ has the Lipschitz extension property for $m \\leq n$ where $\\mathbb{H}^n$ is the sub-Riemannian Heisenberg group. That is, for some $C>0$, any $L$-Lipschitz map from a subset of $\\mathbb{R}^m$ into $\\mathbb{H}^n$ can be extended to a $CL$-Lipschitz mapping on $\\mathbb{R}^m$. In this paper, we construct Sobolev extensions of such Lipschitz mappings with no restriction on the dimension $m$. We prove that any Lipschitz mapping from a compact subset of $\\mathbb{R}^m$ into $\\mathbb{H}^n$ may be extended to a Sobolev mapping on any bounded domain containing the set. This result is then generalized to include mappings into any Lipschitz $(n-1)$-connected metric space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-02T15:02:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q15",
      "53C17",
      "46E35",
      "54C20"
    ],
    "keywords": [
      "lipschitz map",
      "sub-riemannian heisenberg group",
      "lipschitz extension property",
      "construct sobolev extensions",
      "compact subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}