{
  "id": "1903.10365",
  "version": "v1",
  "published": "2019-03-25T14:31:56.000Z",
  "updated": "2019-03-25T14:31:56.000Z",
  "title": "Green's functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz'ya inequalities on half spaces",
  "authors": [
    "Guozhen Lu",
    "Qiaohua Yang"
  ],
  "comment": "33 pages",
  "categories": [
    "math.CA",
    "math.AP",
    "math.DG"
  ],
  "abstract": "Using the Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the $\\frac{n-1}{2}$-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension $n$ coincides with the best $\\frac{n-1}{2}$-th order Sobolev constant when $n$ is odd and $n\\geq9$ (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the $k-$th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of dimension $n$ and $k$-th order derivatives (see Theorem 1.7). Precise expressions and optimal bounds for Green's functions of the operator $ -\\Delta_{\\mathbb{H}}-\\frac{(n-1)^{2}}{4}$ on the hyperbolic space $\\mathbb{B}^n$ and operators of the product form are given, where $\\frac{(n-1)^{2}}{4}$ is the spectral gap for the Laplacian $-\\Delta_{\\mathbb{H}}$ on $\\mathbb{B}^n$. Finally, we give the precise expression and optimal pointwise bound of Green's function of the Paneitz and GJMS operators on hyperbolic space, which are of their independent interest (see Theorem 1.10).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-25T14:31:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "greens function",
      "hyperbolic space",
      "sharp hardy-sobolev-mazya inequalities",
      "gjms operators",
      "th order hardy-sobolev-mazya inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}