{
  "id": "1910.13747",
  "version": "v1",
  "published": "2019-10-30T10:01:26.000Z",
  "updated": "2019-10-30T10:01:26.000Z",
  "title": "A square function involving the center of mass and rectifiability",
  "authors": [
    "Michele Villa"
  ],
  "comment": "32 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "For a Radon measure $\\mu$ on $\\mathbb{R}^d$, define $C^n_\\mu(x, t)= \\ (\\frac{1}{t^n} \\ |\\int_{B(x,t)} \\frac{x-y}{t} \\, d\\mu(y)\\ | \\ )$. This coefficient quantifies how symmetric the measure $\\mu$ is by comparing the center of mass at a given scale and location to the actual center of the ball. We show that if $\\mu$ is $n$-rectifiable, then $ \\int_0^\\infty |C^n_\\mu(x,t)|^2 \\frac{dt}{t} < \\infty \\, \\, \\mu\\mbox{-almost everywhere}. $ Together with a previous result of Mayboroda and Volberg, where they showed that the converse holds true, this gives a characterisation of $n$-rectifiability. To prove our main result, we also show that for an $n$-uniformly rectifiable measure, $|C_\\mu^n(x,t)|^2 dt/t d\\mu$ is a Carleson measure on $\\mathrm{spt}(\\mu) \\times (0,\\infty)$. We also show that, whenever a measure $\\mu$ is $1$-rectifiable in the plane, then the same Dini condition as above holds for more general kernels. Moreover, we give a characterisation of uniform 1-rectifiability in the plane in terms of a Carleson measure condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-30T10:01:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "28A12",
      "28A78"
    ],
    "keywords": [
      "square function",
      "rectifiability",
      "carleson measure condition",
      "converse holds true",
      "coefficient quantifies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}