{
  "id": "2405.09823",
  "version": "v1",
  "published": "2024-05-16T05:49:18.000Z",
  "updated": "2024-05-16T05:49:18.000Z",
  "title": "Boundary Hardy inequality on functions of bounded variation",
  "authors": [
    "Adimurthi",
    "Prosenjit Roy",
    "Vivek Sahu"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \\infty, \\ ~\\Omega$ is bounded Lipschitz domain, then for all $u \\in C^{\\infty}_{c}(\\Omega)$, $$\\int_{\\Omega} \\frac{|u(x)|^{p}}{\\delta^{p}_{\\Omega}(x)} dx \\leq C\\int_{\\Omega} |\\nabla u(x) |^{p}dx,$$ where $\\delta_\\Omega(x)$ is the distance function from $\\Omega^c$. In this article, we address the long standing open question on the case $p=1$ by establishing appropriate boundary Hardy inequalities in the space of functions of bounded variation. We first establish appropriate inequalities on fractional Sobolev spaces $W^{s,1}(\\Omega)$ and then Brezis, Bourgain and Mironescu's result on limiting behavior of fractional Sobolev spaces as $s\\rightarrow 1^{-}$ plays an important role in the proof. Moreover, we also derive an infinite series Hardy inequality for the case $p=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-16T05:49:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "26D15",
      "39B62"
    ],
    "keywords": [
      "bounded variation",
      "fractional sobolev spaces",
      "infinite series hardy inequality",
      "establishing appropriate boundary hardy inequalities",
      "classical boundary hardy inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}