{
  "id": "1406.4769",
  "version": "v2",
  "published": "2014-06-18T15:35:59.000Z",
  "updated": "2014-11-14T11:38:34.000Z",
  "title": "A T(P) theorem for Sobolev spaces on domains",
  "authors": [
    "Martí Prats",
    "Xavier Tolsa"
  ],
  "comment": "35 pages, 6 figures",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given $0<s\\leq1$, $1<p<\\infty$ with $sp>2$ and a Lipschitz domain $\\Omega\\subset \\mathbb{C}$, the Beurling transform $Bf=- {\\rm p.v.}\\frac1{\\pi z^2}*f$ is bounded in the Sobolev space $W^{s,p}(\\Omega)$ if and only if $B\\chi_\\Omega\\in W^{s,p}(\\Omega)$. In this paper we obtain a generalized version of the former result valid for any $s\\in \\mathbb{N}$ and for a larger family of Calder\\'on-Zygmund operators in any ambient space $\\mathbb{R}^d$ as long as $p>d$. In that case we need to check the boundedness not only over the characteristic function of the domain, but over a finite collection of polynomials restricted to the domain. Finally we find a sufficient condition in terms of Carleson measures for $p\\leq d$. In the particular case $s=1$, this condition is in fact necessary, which yields a complete characterization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-18T15:35:59.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-14T11:38:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sobolev space",
      "beurling transform",
      "result valid",
      "calderon-zygmund operators",
      "lipschitz domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.4769P"
    }
  }
}