{
  "id": "1506.00025",
  "version": "v1",
  "published": "2015-05-29T20:47:53.000Z",
  "updated": "2015-05-29T20:47:53.000Z",
  "title": "The Dubovitskiĭ-Sard Theorem in Sobolev Spaces",
  "authors": [
    "Piotr Hajłasz",
    "Scott Zimmerman"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "The Sard theorem from 1942 requires that a mapping $f:\\mathbb{R}^n \\to \\mathbb{R}^m$ is of class $C^k$, $k > \\max (n-m,0)$. In 1957 Duvovitski\\u{\\i} generalized Sard's theorem to the case of $C^k$ mappings for all $k$. Namely he proved that, for almost all $y\\in \\mathbb{R}^m$, $\\mathcal{H}^{\\ell}(C_f \\cap f^{-1}(y))=0$ where $\\ell = \\max(n-m-k+1,0)$, ${\\mathcal H}^{\\ell}$ denotes the Hausdorff measure, and $C_f$ is the set of critical points of $f$. In 2001 De Pascale proved that the Sard theorem holds true for Sobolev mappings of the class $W_{\\rm loc}^{k,p}(\\mathbb{R}^n,\\mathbb{R}^m)$, $k>\\max(n-m,0)$ and $p>n$. We will show that also Dubovitski\\u{\\i}'s theorem can be generalized to the case of $W_{\\rm loc}^{k,p}(\\mathbb{R}^n,\\mathbb{R}^m)$ mappings for all $k\\in\\mathbb{N}$ and $p>n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-29T20:47:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "58C25"
    ],
    "keywords": [
      "sobolev spaces",
      "dubovitskiĭ-sard theorem",
      "sard theorem holds true",
      "generalized sards theorem",
      "sobolev mappings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}