{
  "id": "1201.5385",
  "version": "v1",
  "published": "2012-01-25T21:04:40.000Z",
  "updated": "2012-01-25T21:04:40.000Z",
  "title": "Smoothness of the Beurling transform in Lipschitz domains",
  "authors": [
    "Victor Cruz",
    "Xavier Tolsa"
  ],
  "comment": "32 pages",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "Let D be a planar Lipschitz domain and consider the Beurling transform of the characteristic function of D, B(1_D). Let 1<p<\\infty and 0<a<1 with ap>1. In this paper we show that if the outward unit normal N on bD, the boundary of D, belongs to the Besov space B_{p,p}^{a-1/p}(bD), then the Beurling transform of 1_D is in the Sobolev space W^{a,p}(D). This result is sharp. Further, together with recent results by Cruz, Mateu and Orobitg, this implies that the Beurling transform is bounded in W^{a,p}(D) if N belongs to B_{p,p}^{a-1/p}(bD), assuming that ap>2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-25T21:04:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "30C62",
      "42B20"
    ],
    "keywords": [
      "beurling transform",
      "smoothness",
      "planar lipschitz domain",
      "outward unit normal",
      "besov space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.5385C"
    }
  }
}