{
  "id": "1602.04729",
  "version": "v1",
  "published": "2016-02-15T16:31:03.000Z",
  "updated": "2016-02-15T16:31:03.000Z",
  "title": "Volterra operators on Hardy spaces of Dirichlet series",
  "authors": [
    "Ole Fredrik Brevig",
    "Karl-Mikael Perfekt",
    "Kristian Seip"
  ],
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "For a Dirichlet series symbol $g(s) = \\sum_{n \\geq 1} b_n n^{-s}$, the associated Volterra operator $\\mathbf{T}_g$ acting on a Dirichlet series $f(s)=\\sum_{n\\ge 1} a_n n^{-s}$ is defined by the integral $f\\mapsto -\\int_{s}^{+\\infty} f(w)g'(w)\\,dw$. We show that $\\mathbf{T}_g$ is a bounded operator on the Hardy space $\\mathcal{H}^p$ of Dirichlet series with $0 < p < \\infty$ if and only if the symbol $g$ satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of ${\\operatorname{BMOA}}(\\mathbb{D})$. A further analogy with classical ${\\operatorname{BMO}}$ is that $\\exp(c|g|)$ is integrable (on the infinite polytorus) for some $c > 0$ whenever $\\mathbf{T}_g$ is bounded. In particular, such $g$ belong to $\\mathcal{H}^p$ for every $p < \\infty$. We relate the boundedness of $\\mathbf{T}_g$ to several other ${\\operatorname{BMO}}$ type spaces: ${\\operatorname{BMOA}}$ in half-planes, the dual of $\\mathcal{H}^1$, and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for $m$-homogeneous symbols as well as for general symbols. Finally, we consider the action of $\\mathbf{T}_g$ on reproducing kernels for appropriate sequences of subspaces of $\\mathcal{H}^2$. Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols $g$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-15T16:31:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B10",
      "30H10",
      "30B50"
    ],
    "keywords": [
      "hardy space",
      "volterra operator",
      "standard carleson measure characterization",
      "number theoretic arguments",
      "dirichlet series symbol"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160204729B"
    }
  }
}