{
  "id": "1303.3159",
  "version": "v1",
  "published": "2013-03-13T13:46:10.000Z",
  "updated": "2013-03-13T13:46:10.000Z",
  "title": "Square functions and spectral multipliers for Bessel operators in UMD spaces",
  "authors": [
    "Jorge J. Betancor",
    "Alejandro J. Castro",
    "Lourdes Rodriguez-Mesa"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper we consider square functions (also called Littlewood-Paley g-functions) associated to Hankel convolutions acting on functions in the Bochner-Lebesgue space $L^p((0,\\infty),B)$, where $B$ is a UMD Banach space. As special cases we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operator $\\Delta_\\lambda=-x^{-\\lambda}\\frac{d}{dx}x^{2\\lambda}\\frac{d}{dx}x^{-\\lambda}$, $\\lambda >0$. We characterize the UMD property for a Banach space $B$ by using $L^p((0,\\infty),B)$-boundedness properties of g-functions defined by Bessel-Poisson semigroups. As a by product we prove that the fact that the imaginary power $\\Delta_\\lambda ^{iw}$, $w\\in \\mathbb{R}\\setminus\\{0\\}$, of the Bessel operator $\\Delta_\\lambda$ is bounded in $L^p ((0,\\infty),B)$, $1<p<\\infty$, characterizes the UMD property for the Banach space $B$. As applications of our results for square functions we establish the boundedness in $L^p((0,\\infty),B)$ of spectral multipliers $m(\\Delta_\\lambda)$ of Bessel operators defined by functions $m$ which are holomorphic in sectors $\\Sigma_\\vartheta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-13T13:46:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "42B15",
      "42B20",
      "46B20",
      "46E40"
    ],
    "keywords": [
      "bessel operator",
      "spectral multipliers",
      "umd spaces",
      "umd property",
      "umd banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.3159B"
    }
  }
}