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Riesz projection and bounded mean oscillation for Dirichlet series

Volume 262 / 2022

Sergei Konyagin, Hervé Queffélec, Eero Saksman, Kristian Seip Studia Mathematica 262 (2022), 121-149 MSC: Primary 30B50; Secondary 42B05, 42B30, 30H30, 30H35. DOI: 10.4064/sm200601-22-5 Published online: 11 August 2021


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 $p \gt 2$. 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 \lt p \lt \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 $\log \log N$.


  • Sergei KonyaginSteklov Institute of Mathematics
    8 Gubkin Street, Moscow
    119991, Russia
  • Hervé QueffélecUniversité Lille Nord de France, USTL
    Laboratoire Paul Painlevé, UMR CNRS 8524
    F-59655 Villeneuve-d’Ascq Cedex, France
  • Eero SaksmanDepartment of Mathematics
    and Statistics
    University of Helsinki
    FI-00170 Helsinki, Finland
  • Kristian SeipDepartment of Mathematical Sciences
    Norwegian University of Science and Technology
    NO-7491 Trondheim, Norway

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