{
  "id": "1803.07975",
  "version": "v1",
  "published": "2018-03-21T15:57:06.000Z",
  "updated": "2018-03-21T15:57:06.000Z",
  "title": "A geometric characterization of the weak-$A_\\infty$ condition for harmonic measure",
  "authors": [
    "Jonas Azzam",
    "Mihalis Mourgoglou",
    "Xavier Tolsa"
  ],
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "Let $\\Omega\\subset\\mathbb R^{n+1}$ be an open set with $n$-AD-regular boundary. In this paper we prove that if the harmonic measure for $\\Omega$ satisfies the so-called weak-$A_\\infty$ condition, then $\\Omega$ satisfies a suitable connectivity condition, namely the weak local John condition. This yields the first geometric characterization of the weak-$A_\\infty$ condition of harmonic measure, which is important because its connection with the Dirichlet problem for the Laplace equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-21T15:57:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B15",
      "28A75",
      "28A78",
      "35J15",
      "35J08"
    ],
    "keywords": [
      "harmonic measure",
      "weak local john condition",
      "first geometric characterization",
      "ad-regular boundary",
      "laplace equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}