{
  "id": "1709.09608",
  "version": "v1",
  "published": "2017-09-27T16:31:04.000Z",
  "updated": "2017-09-27T16:31:04.000Z",
  "title": "Improved Moser-Trudinger type inequalities in the hyperbolic space $\\mathbb H^n$",
  "authors": [
    "Van Hoang Nguyen"
  ],
  "comment": "14 pages, comment are welcome",
  "categories": [
    "math.FA",
    "math.AP"
  ],
  "abstract": "We establish an improved version of the Moser-Trudinger inequality in the hyperbolic space $\\mathbb H^n$, $n\\geq 2$. Namely, we prove the following result: for any $0 \\leq \\lambda < \\left(\\frac{n-1}n\\right)^n$, then we have $$ \\sup_{\\substack{u\\in C_0^\\infty(\\mathbb H^n)\\\\ \\int_{\\mathbb H^n} |\\nabla_g u|_g^n d\\text{Vol}_g -\\lambda \\int_{\\mathbb H^n} |u|^n d\\text{ Vol}_g \\leq 1}} \\int_{\\mathbb H^n} \\Phi_n(\\alpha_n |u|^{\\frac{n}{n-1}}) d\\text{ Vol}_g < \\infty, $$ where $\\alpha_n = n \\omega_{n-1}^{\\frac1{n-1}}$, $\\omega_{n-1}$ denotes the surface area of the unit sphere in $\\mathbb R^n$ and $\\Phi_n(t) = e^t -\\sum_{j=0}^{n-2}\\frac{t^j}{j!}$. This improves the Moser-Trudinger inequality in hyperbolic spaces obtained recently by Mancini and Sandeep, by Mancini, Sandeep and Tintarev and by Adimurthi and Tintarev. In the limiting case $\\lambda =(\\frac{n-1}n)^n$, we prove a Moser-Trudinger inequality with exact growth in $\\mathbb H^n$, $$ \\sup_{\\substack{u\\in C_0^\\infty(\\mathbb H^n)\\\\ \\int_{\\mathbb H^n} |\\nabla_g u|_g^n d\\text{ Vol}_g -(\\frac{n-1}n)^n \\int_{\\mathbb H^n} |u|^n d\\text{ Vol}_g \\leq 1}} \\frac{1}{\\int_{\\mathbb H^n} |u|^n d\\text{ Vol}_g}\\int_{\\mathbb H^n} \\frac{\\Phi_n(\\alpha_n |u|^{\\frac{n}{n-1}})}{(1+ |u|)^{\\frac n{n-1}}} d\\text{ Vol}_g < \\infty. $$ This improves the Moser-Trudinger inequality with exact growth in $\\mathbb H^n$ established by Lu and Tang.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-27T16:31:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D10",
      "46E35"
    ],
    "keywords": [
      "hyperbolic space",
      "moser-trudinger type inequalities",
      "moser-trudinger inequality",
      "exact growth",
      "unit sphere"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}