{
  "id": "1201.5403",
  "version": "v1",
  "published": "2012-01-25T22:14:44.000Z",
  "updated": "2012-01-25T22:14:44.000Z",
  "title": "Regularity of C^1 and Lipschitz domains in terms of the Beurling transform",
  "authors": [
    "Xavier Tolsa"
  ],
  "comment": "35 pages",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "Let D be a bounded planar C^1 domain, or a Lipschitz domain \"flat enough\", and consider the Beurling transform of 1_D, the characteristic function of D. Using a priori estimates, in this paper we solve the following free boundary problem: if the Beurling transform of 1_D belongs to the Sobolev space W^{a,p}(D) for 0<a\\leq 1, 1<p<\\infty such that ap>1, then the outward unit normal N on bD, the boundary of D, is in the Besov space B_{p,p}^{a-1/p}(bD). The converse statement, proved previously by Cruz and Tolsa, also holds. So we have that B(1_D) is in W^{a,p}(D) if and only if N is in B_{p,p}^{a-1/p}(bD). Together with recent results by Cruz, Mateu and Orobitg, from the preceding equivalence one infers that the Beurling transform is bounded in W^{a,p}(D) if and only if the outward unit normal N belongs to B_{p,p}^{a-1/p}(bD), assuming that ap>2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-25T22:14:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "35R35",
      "30C62"
    ],
    "keywords": [
      "beurling transform",
      "lipschitz domain",
      "outward unit normal",
      "regularity",
      "free boundary problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}