{
  "id": "1501.01572",
  "version": "v1",
  "published": "2015-01-07T17:47:33.000Z",
  "updated": "2015-01-07T17:47:33.000Z",
  "title": "Characterization of $n$-rectifiability in terms of Jones' square function: Part II",
  "authors": [
    "Jonas Azzam",
    "Xavier Tolsa"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "We show that a Radon measure $\\mu$ in $\\mathbb R^d$ which is absolutely continuous with respect to the $n$-dimensional Hausdorff measure $H^n$ is $n$-rectifiable if the so called Jones' square function is finite $\\mu$-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all $n$-rectifiable measures which are absolutely continuous with respect to $H^{n}$. Further, in this paper we also investigate the relationship between the Jones' square function and the so called Menger curvature of a measure with linear growth.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-07T17:47:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "28A78",
      "42B20"
    ],
    "keywords": [
      "square function",
      "characterization",
      "rectifiability",
      "dimensional hausdorff measure",
      "radon measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150101572A"
    }
  }
}