{
  "id": "1804.02277",
  "version": "v1",
  "published": "2018-04-04T16:58:06.000Z",
  "updated": "2018-04-04T16:58:06.000Z",
  "title": "Remarks on Generalized Hardy Algebras",
  "authors": [
    "Romeo Meštrović",
    "Žarko Pavićević",
    "Novo Labudović"
  ],
  "comment": "18 pages, no figures, Journal-ref: Mathematica Montisnigri, vol. 11 (1999), pp. 25-42; Mathematical Reviews: MR1781340 (2001h:46040)",
  "journal": "Mathematica Montisnigri 11 (1999), 25-42",
  "categories": [
    "math.FA"
  ],
  "abstract": "For a measure space $(\\Omega, \\Sigma, \\mu)$ with a positive finite measure $\\mu$, and a positive real number $p$, we define the space $L_p^{+}(\\mu)=L_p^{+}$ of all (equivalence classes of) $\\Sigma$-measurable complex functions $f$ defined on $\\Omega$ such that the function $\\left(\\log^+|f|\\right)^p$ is integrable with respect to $\\mu $.We define the metric $d_p$ on $L^{+}_p$ which generalizes the metric introduced by Gamelin and Lumer in [G] for the case $p=1$. It is shown that the space $L^{+}_p$ is a topological algebra. On the other hand, one can define on the space $L_p^{+}$ an equivalent $F$-norm $| \\cdot|_p$ that makes $L_p^{+}$ into an Orlicz space. For the case of the normalized Lebesgue's measure $dt/2\\pi$ on $[0,2\\pi)$, it follows that the class $N^p(1<p<\\infty)$ introduced by I. I. Privalov in [P], may be considered as a generalization of the Smirnov class $N^+$. Furthermore, $N^p(1<p<\\infty)$ with the associated modular becomes an Hardy-Orlicz class. Finally, for a strictly positive and measurable on $[0,2\\pi)$ function $w$, we define the generalized Orlicz space $L_p^{w}(\\mathrm{d}t/2\\pi)=L^w_p$ with the modular $\\rho^w_p$ given by the function $\\psi_w(t,u)=\\big(\\log(1+uw(t))\\big)^p$, with a \"weight\" $w$. We observe that the space $L^w_p$ is a generalized Orlicz space with respect to the modular $\\rho^w_p$. We examine and compare different topologies induced on $L^w_p$ by corresponding \"weights\" $w$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-04T16:58:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E30",
      "30H05",
      "46J15"
    ],
    "keywords": [
      "generalized hardy algebras",
      "generalized orlicz space",
      "positive finite measure",
      "positive real number",
      "measure space"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}