{
  "id": "1010.0461",
  "version": "v3",
  "published": "2010-10-04T01:13:06.000Z",
  "updated": "2011-07-31T14:14:42.000Z",
  "title": "New upper bounds for the constants in the Bohnenblust-Hille inequality",
  "authors": [
    "Daniel Pellegrino",
    "Juan B. Seoane-Sepúlveda"
  ],
  "comment": "The present version improves the constants of the previous version",
  "categories": [
    "math.FA"
  ],
  "abstract": "A classical inequality due to Bohnenblust and Hille states that for every positive integer $m$ there is a constant $C_{m}>0$ so that $$(\\sum\\limits_{i_{1},...,i_{m}=1}^{N}|U(e_{i_{^{1}}},...,e_{i_{m}})| ^{\\frac{2m}{m+1}}) ^{\\frac{m+1}{2m}}\\leq C_{m}| U|$$ for every positive integer $N$ and every $m$-linear mapping $U:\\ell_{\\infty}^{N}\\times...\\times\\ell_{\\infty}^{N}\\rightarrow\\mathbb{C}$, where $C_{m}=m^{\\frac{m+1}{2m}}2^{\\frac{m-1}{2}}.$ The value of $C_{m}$ was improved to $C_{m}=2^{\\frac{m-1}{2}}$ by S. Kaijser and more recently H. Qu\\'{e}ffelec and A. Defant and P. Sevilla-Peris remarked that $C_{m}=(\\frac{2}{\\sqrt{\\pi}})^{m-1}$ also works. The Bohnenblust--Hille inequality also holds for real Banach spaces with the constants $C_{m}=2^{\\frac{m-1}{2}}$. In this note we show that a recent new proof of the Bohnenblust--Hille inequality (due to Defant, Popa and Schwarting) provides, in fact, quite better estimates for $C_{m}$ for all values of $m \\in \\mathbb{N}$. In particular, we will also show that, for real scalars, if $m$ is even with $2\\leq m\\leq 24$, then $$C_{\\mathbb{R},m}=2^{1/2}C_{\\mathbb{R},m/2}.$$ We will mainly work on a paper by Defant, Popa and Schwarting, giving some remarks about their work and explaining how to, numerically, improve the previously mentioned constants.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-07-31T14:14:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bohnenblust-hille inequality",
      "upper bounds",
      "positive integer",
      "real banach spaces",
      "quite better estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.0461P"
    }
  }
}