{
  "id": "1808.08882",
  "version": "v1",
  "published": "2018-08-27T15:28:58.000Z",
  "updated": "2018-08-27T15:28:58.000Z",
  "title": "Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one",
  "authors": [
    "Guy David",
    "Max Engelstein",
    "Svitlana Mayboroda"
  ],
  "comment": "36 pages. Comments welcome",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "In this paper, we characterize the rectifiability (both uniform and not) of an Ahlfors regular set, E, of arbitrary co-dimension by the behavior of a regularized distance function in the complement of that set. In particular, we establish a certain version of the Riesz transform characterization of rectifiability for lower-dimensional sets. We also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of E. This allows us to precisely compute the harmonic measure of those sets associated to this degenerate operator and prove that, in a sharp contrast with the usual setting of co-dimension one, a converse to the Dahlberg's theorem (see [Da] and [DFM2]) must be false on lower dimensional boundaries without additional assumptions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-27T15:28:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35"
    ],
    "keywords": [
      "harmonic measure",
      "non-tangential limits",
      "co-dimensions larger",
      "square functions",
      "ahlfors regular set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}