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## Riesz projection and bounded mean oscillation for Dirichlet series

Studia Mathematica MSC: Primary 30B50; Secondary 42B05, 42B30, 30H30, 30H35. DOI: 10.4064/sm200601-22-5 Opublikowany online: 11 August 2021

#### Streszczenie

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 $\log \log N$.

#### Autorzy

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

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