{
  "id": "1607.00418",
  "version": "v1",
  "published": "2016-07-01T21:48:21.000Z",
  "updated": "2016-07-01T21:48:21.000Z",
  "title": "BMO solvability and absolute continuity of harmonic measure",
  "authors": [
    "Steve Hofmann",
    "Phi Le"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We show that for a uniformly elliptic divergence form operator $L$, defined in an open set $\\Omega$ with Ahlfors-David regular boundary, BMO-solvability implies scale invariant quantitative absolute continuity (the weak-$A_\\infty$ property) of elliptic-harmonic measure with respect to surface measure on $\\partial \\Omega$. We do not impose any connectivity hypothesis, qualitative or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of $\\Omega$. In this generality, our results are new even for the Laplacian. Moreover, we obtain a converse, under the additional assumption that $\\Omega$ satisfies an interior Corkscrew condition, in the special case that $L$ is the Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-01T21:48:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J08",
      "42B25",
      "42B37"
    ],
    "keywords": [
      "harmonic measure",
      "bmo solvability",
      "invariant quantitative absolute continuity",
      "uniformly elliptic divergence form operator",
      "implies scale invariant quantitative absolute"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}