{
  "id": "1408.6979",
  "version": "v1",
  "published": "2014-08-29T10:59:30.000Z",
  "updated": "2014-08-29T10:59:30.000Z",
  "title": "Rectifiable measures, square functions involving densities, and the Cauchy transform",
  "authors": [
    "Xavier Tolsa"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "This paper is devoted to the proof of two related results. The first one asserts that if $\\mu$ is a Radon measure in $\\mathbb R^d$ satisfying $$\\limsup_{r\\to 0} \\frac{\\mu(B(x,r))}{r}>0\\quad \\text{ and }\\quad \\int_0^1\\left|\\frac{\\mu(B(x,r))}{r} - \\frac{\\mu(B(x,2r))}{2r}\\right|^2\\,\\frac{dr}r< \\infty$$ for $\\mu$-a.e. $x\\in\\mathbb R^d$, then $\\mu$ is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set $E\\subset\\mathbb R^d$ with finite $1$-dimensional Hausdorff measure $H^1$ is rectifiable if and only $$\\int_0^1\\left|\\frac{H^1(E\\cap B(x,r))}{r} - \\frac{H^1(E\\cap B(x,2r))}{2r}\\right|^2\\,\\frac{dr}r< \\infty \\quad\\mbox{ for $H^1$-a.e. $x\\in E$.}$$ The second result of the paper deals with the relationship between a similar square function in the complex plane and the Cauchy transform $C_\\mu f(z) = \\int \\frac1{z-\\xi}\\,f(\\xi)\\,d\\mu(\\xi)$. Suppose that $\\mu$ has linear growth, that is, $\\mu(B(z,r))\\leq c\\,r$ for all $z\\in\\mathbb C$ and all $r>0$. It is proved that $C_\\mu$ is bounded in $L^2(\\mu)$ if and only if $$ \\int_{z\\in Q}\\int_0^\\infty\\left|\\frac{\\mu(Q\\cap B(z,r))}{r} - \\frac{\\mu(Q\\cap B(z,2r))}{2r}\\right|^2\\,\\frac{dr}r\\,d\\mu(z)\\leq c\\,\\mu(Q) \\quad\\mbox{ for every square $Q\\subset\\mathbb C$.} $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-29T10:59:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "42B20"
    ],
    "keywords": [
      "cauchy transform",
      "rectifiable measures",
      "dimensional hausdorff measure",
      "similar square function",
      "radon measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.6979T"
    }
  }
}