{
  "id": "1307.0804",
  "version": "v2",
  "published": "2013-07-02T19:39:17.000Z",
  "updated": "2014-09-15T13:51:55.000Z",
  "title": "Multiscale analysis of 1-rectifiable measures: necessary conditions",
  "authors": [
    "Matthew Badger",
    "Raanan Schul"
  ],
  "comment": "16 pages, v2: expanded introduction, improved Lemma 2.6 (now 2.7), corrected mistake in proof of Proposition 3.1",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\\Bbb{R}^n$, $n\\geq 2$. To each locally finite Borel measure $\\mu$, we associate a function $\\widetilde J_2(\\mu, x)$ which uses a weighted sum to record how closely the mass of $\\mu$ is concentrated on a line in the triples of dyadic cubes containing $x$. We show that $\\widetilde J_2(\\mu, x) < \\infty$ $\\mu$-a.e. is a necessary condition for $\\mu$ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result is that no assumption is made on the upper Hausdorff density of the measure. Thus we are able to analyze generic 1-rectifiable measures that are mutually singular with the 1-dimensional Hausdorff measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-02T19:39:17.000Z",
      "abstract": "We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in R^n, n >= 2. To each locally finite Borel measure mu, we associate a function ~J_2(mu,x) which uses a weighted sum to record how closely the mass of mu is concentrated on a line in the triples of dyadic cubes containing x. We show that ~J_2(mu,x) < infinity mu-a.e. is a necessary condition for \\mu to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result is that no assumption is made on the upper Hausdorff density of the measure. Thus we are able to analyze generic 1-rectifiable measures that are mutually singular with the 1-dimensional Hausdorff measure.",
      "comment": "14 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-15T13:51:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75"
    ],
    "keywords": [
      "necessary condition",
      "multiscale analysis",
      "locally finite borel measure mu",
      "upper hausdorff density",
      "hausdorff measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.0804B"
    }
  }
}