{
  "id": "1812.11419",
  "version": "v1",
  "published": "2018-12-29T18:29:42.000Z",
  "updated": "2018-12-29T18:29:42.000Z",
  "title": "Capacitary differentiability of potentials of finite Radon measures",
  "authors": [
    "Joan Verdera"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We study differentiability properties of a potential of the type $K\\star \\mu$, where $\\mu$ is a finite Radon measure in $\\mathbb{R}^N$ and the kernel $K$ satisfies $|\\nabla^j K(x)| \\le C\\, |x|^{-(N-1+j)}, \\quad j=0,1,2.$ We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vall\\'ee Poussin sense associated with the kernel $|x|^{-(N-1)}.$ We require that the first order remainder at a point is small when measured by means of a normalized weak capacity \"norm\" in balls of small radii centered at the point. This implies weak $L^{N/(N-1)}$ differentiability and thus $L^{p}$ differentiability in the Calder\\'on--Zygmund sense for $1\\le p < N/(N-1)$. We show that $K\\star \\mu$ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for $K\\star \\mu.$ As an application, we study level sets of newtonian potentials of finite Radon measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-29T18:29:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20"
    ],
    "keywords": [
      "finite radon measure",
      "capacitary differentiability",
      "capacity sense",
      "study level sets",
      "study differentiability properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}