{
  "id": "2006.11046",
  "version": "v1",
  "published": "2020-06-19T10:03:02.000Z",
  "updated": "2020-06-19T10:03:02.000Z",
  "title": "The precise representative for the gradient of the Riesz potential of a finite measure",
  "authors": [
    "Julià Cufí",
    "Augusto C. Ponce",
    "Joan Verdera"
  ],
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Given a finite nonnegative Borel measure $m$ in $\\mathbb{R}^{d}$, we identify the Lebesgue set $\\mathcal{L}(V_{s}) \\subset \\mathbb{R}^{d}$ of the vector-valued function $$V_{s}(x) = \\int_{\\mathbb{R}^{d}}\\frac{x - y}{|x - y|^{s + 1}} \\mathrm{d}m(y), $$ for any order $0 < s < d$. We prove that $a \\in \\mathcal{L}(V_{s})$ if and only if the integral above has a principal value at $a$ and $$\\lim_{r \\to 0}{\\frac{m(B_{r}(a))}{r^{s}}} = 0.$$ In that case, the precise representative of $V_{s}$ at $a$ coincides with the principal value of the integral.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-19T10:03:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B15",
      "26B05"
    ],
    "keywords": [
      "finite measure",
      "precise representative",
      "riesz potential",
      "principal value",
      "finite nonnegative borel measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}