Riesz projection and bounded mean oscillation for Dirichlet series
Volume 262 / 2022
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 $p \gt 2$. 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 $\log \log N$.
