{
  "id": "0907.4188",
  "version": "v2",
  "published": "2009-07-23T23:22:22.000Z",
  "updated": "2012-05-09T05:29:32.000Z",
  "title": "Quasiconformal maps, analytic capacity, and non linear potentials",
  "authors": [
    "Xavier Tolsa",
    "Ignacio Uriarte-Tuero"
  ],
  "comment": "57 pages; typos corrected",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "In this paper we prove that if $\\phi:\\C\\to\\C$ is a $K$-quasiconformal map, with $K>1$, and $E\\subset \\C$ is a compact set contained in a ball $B$, then $$\\frac{\\dot C_{\\frac{2K}{2K+1},\\frac{2K+1}{K+1}}(E)}{\\diam(B)^{\\frac2{K+1}}} \\geq c^{-1} (\\frac{\\gamma(\\phi(E))}{\\diam(\\phi(B))})^{\\frac{2K}{K+1}},$$ where $\\gamma$ stands for the analytic capacity and $\\dot C_{\\frac{2K}{2K+1},\\frac{2K+1}{K+1}}$ is a capacity associated to a non linear Riesz potential. As a consequence, if $E$ is not $K$-removable (i.e. removable for bounded $K$-quasiregular maps), it has positive capacity $\\dot C_{frac{2K}{2K+1},\\frac{2K+1}{K+1}}$. This improves previous results that assert that $E$ must have non $\\sigma$-finite Hausdorff measure of dimension $2/(K+1)$. We also show that the indices $\\frac{2K}{2K+1}$, $\\frac{2K+1}{K+1}$ are sharp, and that Hausdorff gauge functions do not appropriately discriminate which sets are $K$-removable. So essentially we solve the problem of finding sharp \"metric\" conditions for $K$-removability.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-05-09T05:29:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C62",
      "31A15",
      "35J15",
      "28A75",
      "49Q15"
    ],
    "keywords": [
      "non linear potentials",
      "quasiconformal map",
      "analytic capacity",
      "non linear riesz potential",
      "finite hausdorff measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.4188T"
    }
  }
}