{
  "id": "1311.0155",
  "version": "v1",
  "published": "2013-11-01T12:01:28.000Z",
  "updated": "2013-11-01T12:01:28.000Z",
  "title": "Compactness of higher-order Sobolev embeddings",
  "authors": [
    "Lenka Slavíková"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We study higher-order compact Sobolev embeddings on a domain $\\Omega \\subseteq \\mathbb R^n$ endowed with a probability measure $\\nu$ and satisfying certain isoperimetric inequality. Given $m\\in \\mathbb N$, we present a condition on a pair of rearrangement-invariant spaces $X(\\Omega,\\nu)$ and $Y(\\Omega,\\nu)$ which suffices to guarantee a compact embedding of the Sobolev space $V^mX(\\Omega,\\nu)$ into $Y(\\Omega,\\nu)$. The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of $(\\Omega,\\nu)$. We then apply this result to the characterization of higher-order compact Sobolev embeddings on concrete measure spaces, including John domains, Maz'ya classes of Euclidean domains and product probability spaces, whose standard example is the Gauss space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-01T12:01:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "46E30"
    ],
    "keywords": [
      "higher-order sobolev embeddings",
      "compactness",
      "study higher-order compact sobolev embeddings",
      "product probability spaces",
      "concrete measure spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.0155S"
    }
  }
}