{
  "id": "2307.09859",
  "version": "v1",
  "published": "2023-07-19T09:42:28.000Z",
  "updated": "2023-07-19T09:42:28.000Z",
  "title": "A variant of Hilbert's inequality and the norm of the Hilbert Matrix on $K^p$",
  "authors": [
    "Vassilis Daskalogiannis",
    "Petros Galanopoulos",
    "Michael Papadimitrakis"
  ],
  "comment": "14 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We prove the nontrivial variant \\[ \\sum\\limits_{m,n=1}^{\\infty}\\Big(\\frac{n}{m}\\Big)^{\\frac{1}{q}-\\frac{1}{p}}\\frac{a_mb_n}{m+n-1}\\leq\\frac{\\pi}{\\sin\\frac{\\pi}{p}} \\Big( \\sum\\limits_{m=1}^{\\infty}a_m^p\\Big)^{\\frac 1p}\\Big( \\sum\\limits_{n=1}^{\\infty}b_n^q\\Big)^{\\frac 1q} \\] of the well known Hilbert's inequality. Then we use this to determine the exact value $\\frac{\\pi}{\\sin\\frac{\\pi}p}$ of the norm of the Hilbert matrix as an operator acting on the Hardy-Littlewood space $K^p$. This space consists of all functions $f(z)=\\sum\\limits_{m=0}^{\\infty}a_mz^m$ analytic in the unit disc with $\\|f\\|_{K^p}^p=\\sum\\limits_{m=0}^{\\infty}(m+1)^{p-2}|a_m|^p<+\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-19T09:42:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A30",
      "47B37",
      "47B91"
    ],
    "keywords": [
      "hilbert matrix",
      "hilberts inequality",
      "unit disc",
      "nontrivial variant",
      "space consists"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}