{
  "id": "0903.3395",
  "version": "v1",
  "published": "2009-03-19T19:26:38.000Z",
  "updated": "2009-03-19T19:26:38.000Z",
  "title": "Hypercontractivity of the Bohnenblust-Hille inequality for polynomials and multidimensional Bohr radii",
  "authors": [
    "Andreas Defant",
    "Leonhard Frerick"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In 1931 Bohnenblust and Hille proved that for each m-homogeneous polynomial $\\sum_{|\\alpha| = m} a_\\alpha z^\\alpha$ on $\\C^n$ the $\\ell^{\\frac{2m}{m+1}}$-norm of its coefficients is bounded from above by a constant $C_m$ (depending only on the degree $m$) times the sup norm of the polynomial on the polydisc $\\mathbb{D}^n$. We prove that this inequality is hypercontractive in the sense that the optimal constant $C_m$ is $\\leq C^m$ where $C \\geq 1$ is an absolute constant. From this we derive that the Bohr radius $K_n$ of the $n$-dimensional polydisc in $\\mathbb{C}^n$ is up to an absolute constant $\\geq \\sqrt{\\log n/n}$; this result was independently and with a differnt proof discovered by Ortega-Cerd{\\`a}, Ouna\\\"ies and Seip. An alternative approach even allows to prove that the Bohr radius $K_n^p$, $1 \\leq p \\leq \\infty $ of the unit ball of $\\ell_n^p ,$ is asymptotically $ \\geq (\\log n/n) ^{1-1/ \\min (p,2)}$. This shows that the upper bounds for $K_n^p$ given by Boas and Khavinson are optimal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-19T19:26:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bohr radius",
      "multidimensional bohr radii",
      "bohnenblust-hille inequality",
      "polynomial",
      "hypercontractivity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.3395D"
    }
  }
}