{
  "id": "1309.1940",
  "version": "v1",
  "published": "2013-09-08T08:37:32.000Z",
  "updated": "2013-09-08T08:37:32.000Z",
  "title": "Sobolev homeomorphisms and Brennan's conjecture",
  "authors": [
    "Vladimir Gol'dshtein",
    "Alexander Ukhlov"
  ],
  "comment": "8 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\Omega \\subset \\mathbb{R}^n$ be a domain that supports the $p$-Poincar\\'e inequality. Given a homeomorphism $\\varphi \\in L^1_p(\\Omega)$, for $p>n$ we show the domain $\\varphi(\\Omega)$ has finite geodesic diameter. This result has a direct application to Brennan's conjecture and quasiconformal homeomorphisms. {\\bf The Inverse Brennan's conjecture} states that for any simply connected plane domain $\\Omega' \\subset\\mathbb C$ with nonempty boundary and for any conformal homeomorphism $\\varphi$ from the unit disc $\\mathbb{D}$ onto $\\Omega'$ the complex derivative $\\varphi'$ is integrable in the degree $s$, $-2<s<2/3$. If $\\Omega'$ is bounded than $-2<s\\leq 2$. We prove that integrability in the degree $s> 2$ is not possible for domains $\\Omega'$ with infinite geodesic diameter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-08T08:37:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sobolev homeomorphisms",
      "inverse brennans conjecture",
      "infinite geodesic diameter",
      "poincare inequality",
      "direct application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.1940G"
    }
  }
}