{
  "id": "2005.11951",
  "version": "v1",
  "published": "2020-05-25T07:30:29.000Z",
  "updated": "2020-05-25T07:30:29.000Z",
  "title": "Riesz projection and bounded mean oscillation for Dirichlet series",
  "authors": [
    "Sergei Konyagin",
    "Hervé Queffélec",
    "Eero Saksman",
    "Kristian Seip"
  ],
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "We prove that the norm of the Riesz projection from $L^\\infty(\\Bbb{T}^n)$ to $L^p(\\Bbb{T}^n)$ is $1$ for all $n\\ge 1$ only if $p\\le 2$, thus solving a problem posed by Marzo and Seip in 2011. This shows that $H^p(\\Bbb{T}^{\\infty})$ does not contain the dual space of $H^1(\\Bbb{T}^{\\infty})$ for any $p>2$. We then note that the dual of $H^1(\\Bbb{T}^{\\infty})$ contains, via the Bohr lift, the space of Dirichlet series in $\\operatorname{BMOA}$ of the right half-plane. We give several conditions showing how this $\\operatorname{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\\Bbb{T}$, we compute its $L^p$ norm when $1<p<\\infty$, and we use this result to show that the $L^\\infty$ norm of the $N$th partial sum of a bounded Dirichlet series over $d$-smooth numbers is of order $\\log\\log N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-25T07:30:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30B50",
      "42B05",
      "42B30",
      "30H30",
      "30H35"
    ],
    "keywords": [
      "dirichlet series",
      "riesz projection",
      "bounded mean oscillation",
      "partial sum operator",
      "th partial sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}