{
  "id": "2105.09012",
  "version": "v1",
  "published": "2021-05-19T09:26:31.000Z",
  "updated": "2021-05-19T09:26:31.000Z",
  "title": "Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets",
  "authors": [
    "Anders Björn",
    "Jana Björn",
    "Panu Lahti"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincar\\'e inequality. In particular, when restricted to Euclidean spaces, a closed set $E\\subset \\mathbf{R}^n$ with zero Lebesgue measure is shown to be removable for $W^{1,p}(\\mathbf{R}^n \\setminus E)$ if and only if $\\mathbf{R}^n \\setminus E$ supports a $p$-Poincar\\'e inequality as a metric space. When $p>1$, this recovers Koskela's result (Ark. Mat. 37 (1999), 291--304), but for $p=1$, as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces $L^{1,p}$. To be able to include $p=1$, we first study extensions of Newtonian Sobolev functions in the case $p=1$ from a noncomplete space $X$ to its completion $\\widehat{X}$. In these results, $p$-path almost open sets play an important role, and we provide a characterization of them by means of $p$-path open, $p$-quasiopen and $p$-finely open sets. We also show that there are nonmeasurable $p$-path almost open subsets of $\\mathbf{R}^n$, $n \\geq 2$, provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with $L^p$-integrable upper gradients, about $p$-quasiopen, $p$-path and $p$-finely open sets, and about Lebesgue points for $N^{1,1}$-functions, to spaces that only satisfy local assumptions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-19T09:26:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31E05",
      "26D10",
      "30L15",
      "30L99",
      "31C45",
      "46E35",
      "46E36"
    ],
    "keywords": [
      "newtonian sobolev spaces",
      "removable sets",
      "characterization",
      "newtonian sobolev functions",
      "finely open sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}