{
  "id": "2308.10303",
  "version": "v1",
  "published": "2023-08-20T15:59:18.000Z",
  "updated": "2023-08-20T15:59:18.000Z",
  "title": "Improved Hardy inequalities on Riemannian Manifolds",
  "authors": [
    "Kaushik Mohanta",
    "Jagmohan Tyagi"
  ],
  "comment": "Accepted for publication in Complex Variables and Elliptic Equations",
  "doi": "10.1080/17476933.2023.2247998",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We study the following version of Hardy-type inequality on a domain $\\Omega$ in a Riemannian manifold $(M,g)$: $$ \\int{\\Omega}|\\nabla u|_g^p\\rho^\\alpha dV_g \\geq \\left(\\frac{|p-1+\\beta|}{p}\\right)^p\\int{\\Omega}\\frac{|u|^p|\\nabla \\rho|_g^p}{|\\rho|^p}\\rho^\\alpha dV_g +\\int{\\Omega} V|u|^p\\rho^\\alpha dV_g, \\quad \\forall\\ u\\in C_c^\\infty (\\Omega). $$ We provide sufficient conditions on $p, \\alpha, \\beta,\\rho$ and $V$ for which the above inequality holds. This generalizes earlier well-known works on Hardy inequalities on Riemannian manifolds. The functional setup covers a wide variety of particular cases, which are discussed briefly: for example, $\\mathbb{R}^N$ with $p<N$, $\\mathbb{R}^N\\setminus \\{0\\}$ with $p\\geq N$, $\\mathbb{H}^N$, etc.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-20T15:59:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J05",
      "35A23",
      "46E35"
    ],
    "keywords": [
      "riemannian manifold",
      "hardy inequalities",
      "generalizes earlier well-known works",
      "functional setup covers",
      "sufficient conditions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Taylor-Francis"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}