{
  "id": "1701.08929",
  "version": "v1",
  "published": "2017-01-31T06:49:27.000Z",
  "updated": "2017-01-31T06:49:27.000Z",
  "title": "Factorizations and Hardy-Rellich-Type Inequalities",
  "authors": [
    "Fritz Gesztesy",
    "Lance Littlejohn"
  ],
  "comment": "13 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{\\alpha,\\beta} := - \\Delta + \\alpha |x|^{-2} x \\cdot \\nabla + \\beta |x|^{-2}$, $\\alpha, \\beta \\in \\mathbb{R}$, $x \\in \\mathbb{R}^n \\backslash \\{0\\}$, $n \\in \\mathbb{N}$, $n \\geq 2$, and its formal adjoint, denoted by $T_{\\alpha,\\beta}^+$, we show that nonnegativity of $T_{\\alpha,\\beta}^+ T_{\\alpha,\\beta}$ on $C_0^{\\infty}(\\mathbb{R}^n \\backslash \\{0\\})$ implies the fundamental inequality, \\begin{align} \\int_{\\mathbb{R}^n} [(\\Delta f)(x)]^2 \\, d^n x &\\geq [(n - 4) \\alpha - 2 \\beta] \\int_{\\mathbb{R}^n} |x|^{-2} |(\\nabla f)(x)|^2 \\, d^n x \\notag \\\\ & \\quad - \\alpha (\\alpha - 4) \\int_{\\mathbb{R}^n} |x|^{-4} |x \\cdot (\\nabla f)(x)|^2 \\, d^n x \\notag \\\\ & \\quad + \\beta [(n - 4) (\\alpha - 2) - \\beta] \\int_{\\mathbb{R}^n} |x|^{-4} |f(x)|^2 \\, d^n x, \\notag \\end{align} for $f \\in C^{\\infty}_0(\\mathbb{R}^n \\backslash \\{0\\})$. A particular choice of values for $\\alpha$ and $\\beta$ yields known Hardy-Rellich-type inequalities, including the classical Rellich inequality and an inequality due to Schmincke. By locality, these inequalities extend to the situation where $\\mathbb{R}^n$ is replaced by an arbitrary open set $\\Omega \\subseteq \\mathbb{R}^n$ for functions $f \\in C^{\\infty}_0(\\Omega \\backslash \\{0\\})$. Perhaps more importantly, we will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order situations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-31T06:49:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A23",
      "35J30",
      "47A63",
      "47F05"
    ],
    "keywords": [
      "inequality",
      "hardy-rellich-type inequalities",
      "even-order partial differential operators yield",
      "factorizations",
      "dimensional homogeneous scalar differential expressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}