{
  "id": "1708.02304",
  "version": "v1",
  "published": "2017-08-07T21:05:00.000Z",
  "updated": "2017-08-07T21:05:00.000Z",
  "title": "Rectifiability of measures and the $β_p$ coefficients",
  "authors": [
    "Xavier Tolsa"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "In some former works of Azzam and Tolsa it was shown that $n$-rectifiability can be characterized in terms of a square function involving the David-Semmes $\\beta_2$ coefficients. In the present paper we construct some counterexamples which show that a similar characterization does not hold for the $\\beta_p$ coefficients with $p\\neq2$. This is in strong contrast with what happens in the case of uniform $n$-rectifiability. In the second part of this paper we provide an alternative argument for a recent result of Edelen, Naber and Valtorta about the $n$-rectifiability of measures with bounded lower $n$-dimensional density. Our alternative proof follows from a slight variant of the corona decomposition in one of the aforementioned works of Azzam and Tolsa and a suitable approximation argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-07T21:05:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "28A80"
    ],
    "keywords": [
      "rectifiability",
      "coefficients",
      "suitable approximation argument",
      "corona decomposition",
      "square function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}