{
  "id": "1410.2782",
  "version": "v1",
  "published": "2014-10-10T13:43:08.000Z",
  "updated": "2014-10-10T13:43:08.000Z",
  "title": "Sets of absolute continuity for harmonic measure in NTA domains",
  "authors": [
    "Jonas Azzam"
  ],
  "categories": [
    "math.CA",
    "math.AP",
    "math.MG"
  ],
  "abstract": "If $\\Omega\\subseteq\\mathbb{R}^{d+1}$ is an NTA domain with harmonic measure $w$ and $E\\subseteq \\partial\\Omega$ is contained in an Ahlfors regular set, then for all $\\tau>0$ there is $E'\\subseteq E$ $d$-rectifiable with $w(E\\backslash E')<\\tau w(E)$ and $w|_{E'}\\ll \\mathscr{H}^{d}|_{E'}\\ll w|_{E'}$, so in particular, $w|_{E}\\ll \\mathscr{H}^{d}|_{E}$. Moreover, this holds quantitatively in the sense that $w$ obeys an $A_{\\infty}$-type condition with respect to $\\mathscr{H}^{d}$ on $E'$, even though $\\partial\\Omega$ may not even be locally $\\mathscr{H}^{d}$-finite. We also show that if $(\\Omega^{c})^{\\circ}$ is also NTA, and if $E\\subseteq\\partial \\Omega$ is in a Lipschitz image of $\\mathbb{R}^{d}$, then for all $\\tau>0$ there is $E'\\subseteq E$ with $\\mathscr{H}^{d}(E\\backslash E')<\\tau\\mathscr{H}^{d}(E)$, $w|_{E'}\\ll\\mathscr{H}^{d}|_{E'}\\ll w|_{E'}$, and such that a similar $A_{\\infty}$-condition holds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-10T13:43:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31A15",
      "28A75",
      "28A78"
    ],
    "keywords": [
      "harmonic measure",
      "nta domain",
      "absolute continuity",
      "ahlfors regular set",
      "type condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.2782A"
    }
  }
}