{
  "id": "1703.08149",
  "version": "v1",
  "published": "2017-03-23T17:27:40.000Z",
  "updated": "2017-03-23T17:27:40.000Z",
  "title": "Sharp Hardy-Adams inequalities for bi-Laplacian on hyperbolic space of dimension four",
  "authors": [
    "Guozhen Lu",
    "Qiaohua Yang"
  ],
  "comment": "27 pages",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We establish sharp Hardy-Adams inequalities on hyperbolic space $\\mathbb{B}^{4}$ of dimension four. Namely, we will show that for any $\\alpha>0$ there exists a constant $C_{\\alpha}>0$ such that \\[ \\int_{\\mathbb{B}^{4}}(e^{32\\pi^{2} u^{2}}-1-32\\pi^{2} u^{2})dV=16\\int_{\\mathbb{B}^{4}}\\frac{e^{32\\pi^{2} u^{2}}-1-32\\pi^{2} u^{2}}{(1-|x|^{2})^{4}}dx\\leq C_{\\alpha}. \\] for any $u\\in C^{\\infty}_{0}(\\mathbb{B}^{4})$ with \\[ \\int_{\\mathbb{B}^{4}}\\left(-\\Delta_{\\mathbb{H}}-\\frac{9}{4}\\right)(-\\Delta_{\\mathbb{H}}+\\alpha)u\\cdot udV\\leq1. \\] As applications, we obtain a sharpened Adams inequality on hyperbolic space $\\mathbb{B}^{4}$ and an inequality which improves the classical Adams' inequality and the Hardy inequality simultaneously. The later inequality is in the spirit of the Hardy-Trudinger-Moser inequality on a disk in dimension two given by Wang and Ye [37] and on any convex planar domain by the authors [26]. The tools of fractional Laplacian, Fourier transform and the Plancherel formula on hyperbolic spaces and symmetric spaces play an important role in our work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-23T17:27:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperbolic space",
      "bi-laplacian",
      "convex planar domain",
      "symmetric spaces play",
      "establish sharp hardy-adams inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}