{
  "id": "1701.01280",
  "version": "v1",
  "published": "2017-01-05T11:25:17.000Z",
  "updated": "2017-01-05T11:25:17.000Z",
  "title": "On remainders and stability of anisotropic $L^{p}$-weighted Hardy inequalities",
  "authors": [
    "Michael Ruzhansky",
    "Durvudkhan Suragan",
    "Nurgissa Yessirkegenov"
  ],
  "comment": "21 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we obtain remainder estimates with sharp constants for $L^{p}$-weighted Hardy inequalities on homogeneous groups, which is also a new result in the Euclidean setting of $\\mathbb R^{n}$. The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are proved. Moreover, we investigate another improved version of $L^{p}$-weighted Hardy inequalities involving a distance and the critical case $p=Q=2$ of the improved form of the Hardy inequalities on the quasi-ball. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is investigated. We also introduce a new type of Caffarelli-Kohn-Nirenberg inequalities, which are novel also in the isotropic case of $\\mathbb R^{n}$, implying for $0<\\delta<1$ and for all functions $f\\in C_{0}^{\\infty}(\\mathbb R^{n}\\backslash\\{0\\})$, the inequalities $$ \\left\\|\\frac{f}{|x|^{\\frac{n-p}{p}+\\delta}}\\right\\|_{L^{p}(\\mathbb R^{n})}\\leq p^{\\delta} \\left\\|\\frac{\\log|x|}{|x|^{\\frac{n-p}{p}}}\\nabla f\\right\\|^{\\delta}_{L^{p}(\\mathbb R^{n})} \\left\\|\\frac{f}{|x|^{\\frac{n-p}{p}}}\\right\\|^{1-\\delta}_{L^{p}(\\mathbb R^{n})}, \\quad 1<p<\\infty, $$ and $$ \\left\\|\\frac{f}{|x|^{\\frac{n-p}{p}+\\delta}}\\right\\|_{L^{p}(\\mathbb R^{n})}\\leq \\left\\|\\frac{\\nabla f}{|x|^{\\frac{n-2p}{p}}}\\right\\|^{1-\\delta}_{L^{p}(\\mathbb R^{n})} \\left\\|\\frac{f}{|x|^{\\frac{n}{p}}}\\right\\|^{\\delta}_{L^{p}(\\mathbb R^{n})} , \\quad 1<p<\\infty. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-05T11:25:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E30",
      "43A80"
    ],
    "keywords": [
      "weighted hardy inequalities",
      "homogeneous groups",
      "anisotropic",
      "uncertainty type principles",
      "isotropic case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}