{
  "id": "1701.01950",
  "version": "v1",
  "published": "2017-01-08T12:27:39.000Z",
  "updated": "2017-01-08T12:27:39.000Z",
  "title": "A new estimate for the constants of an inequality due to Hardy and Littlewood",
  "authors": [
    "Antonio Gomes Nunes"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "One of the classical Hardy--Littlewood inequalities for $m$-linear forms on $\\ell _{p}$ spaces asserts that \\begin{equation*} \\left( \\sum_{j_{1},...,j_{m}=1}^{\\infty }\\left\\vert T\\left( e_{j_{1}},\\ldots ,e_{j_{m}}\\right) \\right\\vert ^{\\frac{p}{p-m}}\\right) ^{\\frac{p-m}{p}}\\leq 2^{\\frac{m-1}{2}}\\left\\Vert T\\right\\Vert \\end{equation*}% for all continuous $m$-linear forms $T:\\ell _{p}\\times \\cdots \\times \\ell _{p}\\rightarrow \\mathbb{R}$ or $\\mathbb{C}$ when $m<p\\leq 2m.$ The case $m=2$ recovers an important inequality proved by Hardy and Littlewood in 1934. The search of optimal constants for these kind of inequalities is a fashionable subject, sometimes with concrete and unexpected applications in different fields of Mathematics and Physics, like, for instance, Quantum Information Theory. It was recently proved by Albuquerque \\textit{et al.} that $2^{\\frac{% m-1}{2}}$ can be improved to $2^{\\frac{\\left( m-1\\right) \\left( p-m\\right) }{% p}}$. In the present paper we give a step further improving this last estimate to $2^{\\frac{\\left( p-2\\right) \\left( m-1\\right) \\left( p-m\\right) }{p^{2}}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-08T12:27:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear forms",
      "quantum information theory",
      "optimal constants",
      "spaces asserts",
      "classical hardy-littlewood inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}