{
  "id": "1302.2422",
  "version": "v2",
  "published": "2013-02-11T09:25:26.000Z",
  "updated": "2013-02-12T08:05:47.000Z",
  "title": "Operator theoretic differences between Hardy and Dirichlet-type spaces",
  "authors": [
    "José Ángel Peláez",
    "Fernando Pérez-González",
    "Jouni Rättyä"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "For $0<p<\\infty $, the Dirichlet-type space $\\Dp$ consists of those analytic functions $f$ in the unit disc $\\D$ such that $\\int_\\D|f'(z)|\\sp p(1-|z|)^{p-1}\\,dA(z)<\\infty$. Motivated by operator theoretic differences between the Hardy space $H^p$ and $\\Dp$, the integral operator {displaymath} T_g(f)(z)=\\int_{0}^{z}f(\\zeta)\\,g'(\\zeta)\\,d\\zeta,\\quad z\\in\\D, {displaymath} acting from one of these spaces to another is studied. In particular, it is shown, on one hand, that $T_g:\\Dp\\to H^p$ is bounded if and only if $g\\in\\BMOA$ when $0<p\\le 2$, and, on the other hand, that this equivalence is very far from being true if $p>2$. Those symbols $g$ such that $T_g:\\Dp\\to H^q$ is bounded (or compact) when $p<q$ are also characterized. Moreover, the best known sufficient $L^\\infty$-type condition for a positive Borel measure $\\mu$ on $\\D$ to be a $p$-Carleson measures for $\\Dp$, $p>2$, is significantly relaxed, and the established result is shown to be sharp in a very strong sense.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-02-12T08:05:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G10",
      "30H10"
    ],
    "keywords": [
      "operator theoretic differences",
      "dirichlet-type space",
      "unit disc",
      "strong sense",
      "integral operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.2422A"
    }
  }
}