{
  "id": "1709.06802",
  "version": "v1",
  "published": "2017-09-20T10:37:02.000Z",
  "updated": "2017-09-20T10:37:02.000Z",
  "title": "Accessible Parts of Boundary for Simply Connected Domains",
  "authors": [
    "Pekka Koskela",
    "Debanjan Nandi",
    "Artur Nicolau"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "For a bounded simply connected domain $\\Omega\\subset\\mathbb{R}^2$, any point $z\\in\\Omega$ and any $0<\\alpha<1$, we give a lower bound for the $\\alpha$-dimensional Hausdorff content of the set of points in the boundary of $\\Omega$ which can be joined to $z$ by a John curve with John constant depending only on $\\alpha$, in terms of the distance of $z$ to $\\partial\\Omega$. In fact this set in the boundary contains the intersection $\\partial\\Omega_z\\cap\\partial\\Omega$ of the boundary of a John sub-domain $\\Omega_z$ of $\\Omega$, centered at $z$, with the boundary of $\\Omega$. This may be understood as a quantitative version of a result of Makarov. This estimate is then applied to obtain the pointwise version of a weighted Hardy inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-20T10:37:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C35",
      "26D15"
    ],
    "keywords": [
      "simply connected domain",
      "accessible parts",
      "dimensional hausdorff content",
      "john constant",
      "weighted hardy inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}