{
  "id": "1401.1866",
  "version": "v2",
  "published": "2014-01-09T00:39:54.000Z",
  "updated": "2014-08-23T17:51:26.000Z",
  "title": "On Sharp Constants for Dual Segal--Bargmann $L^p$ Spaces",
  "authors": [
    "William E. Gryc",
    "Todd Kemp"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We study dilated holomorphic $L^p$ space of Gaussian measures over $\\mathbb{C}^n$, denoted $\\mathcal{H}_{p,\\alpha}^n$ with variance scaling parameter $\\alpha>0$. The duality relations $(\\mathcal{H}_{p,\\alpha}^n)^\\ast \\cong \\mathcal{H}_{p',\\alpha}$ hold with $\\frac{1}{p}+\\frac{1}{p'}=1$, but not isometrically. We identify the sharp lower constant comparing the norms on $\\mathcal{H}_{p',\\alpha}$ and $(\\mathcal{H}_{p,\\alpha}^n)^\\ast$, and provide upper and lower bounds on the sharp upper constant. We prove several suggestive partial results on the sharpness of the upper constant. One of these partial results leads to a sharp bound on each Taylor coefficient of a function in the Fock space for $n=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-09T00:39:54.000Z",
      "abstract": "We study dilated holomorphic $L^p$ space of Gaussian measures over $\\mathbb{C}^n$, denoted $\\mathcal{H}_{p,\\alpha}^n$ with variance scalaing parameter $\\alpha>0$. The duality relations $(\\mathcal{H}_{p,\\alpha}^n)^\\ast \\cong \\mathcal{H}_{p',\\alpha}$ hold with $\\frac{1}{p}+\\frac{1}{p'}=1$, but not isometrically. We identify the sharp lower constant comparing the norms on $\\mathcal{H}_{p',\\alpha}$ and $(\\mathcal{H}_{p,\\alpha}^n)^\\ast$, and provide upper and lower bounds on the sharp upper constant. We prove a local version of the sharpness of the upper constant.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-08-23T17:51:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B32",
      "46E15",
      "32A37",
      "42B35"
    ],
    "keywords": [
      "dual segal-bargmann",
      "sharp constants",
      "sharp upper constant",
      "variance scalaing parameter",
      "duality relations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.1866G"
    }
  }
}