{
  "id": "1501.01569",
  "version": "v1",
  "published": "2015-01-07T17:38:05.000Z",
  "updated": "2015-01-07T17:38:05.000Z",
  "title": "Characterization of $n$-rectifiability in terms of Jones' square function: Part I",
  "authors": [
    "Xavier Tolsa"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "In this paper it is shown that if $\\mu$ is a finite Radon measure in $\\mathbb R^d$ which is $n$-rectifiable and $1\\leq p\\leq 2$, then $$\\int_0^\\infty \\beta_{\\mu,p}^n(x,r)^2\\,\\frac{dr}r<\\infty \\quad\\mbox{ for $\\mu$-a.e. $x\\in\\mathbb R^d$,}$$ where $$\\beta_{\\mu,p}^n(x,r) = \\inf_L \\left(\\frac1{r^n} \\int_{\\bar B(x,r)} \\left(\\frac{\\mathrm dist(y,L)}{r}\\right)^p\\,d\\mu(y)\\right)^{1/p},$$ with the infimum taken over all the $n$-planes $L\\subset \\mathbb R^d$. The $\\beta_{\\mu,p}^n$ coefficients are the same as the ones considered by David and Semmes in the setting of the so called uniform $n$-rectifiability. An analogous necessary condition for $n$-rectifiability in terms of other coefficients involving some variant of the Wasserstein distance $W_1$ is also proved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-07T17:38:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "28A78",
      "42B20"
    ],
    "keywords": [
      "square function",
      "rectifiability",
      "characterization",
      "finite radon measure",
      "infimum taken"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150101569T"
    }
  }
}