Riesz projection and bounded mean oscillation for Dirichlet series

Abstract

We prove that the norm of the Riesz projection from $L^{\infty }({\mathbb{T}}^n)$ to $L^p({\mathbb{T}}^n)$ 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 $H^p({\mathbb{T}}^{\infty })$ does not contain the dual space of $H^1({\mathbb{T}}^{\infty })$ for any $p>2$. We then note that the dual of $H^1({\mathbb{T}}^{\infty })$ contains, via the Bohr lift, the space of Dirichlet series in $\mathrm{BMOA}$ of the right half-plane. We give several conditions showing how this $\mathrm{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\mathbb{T}$, 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 $\log \log N$.

This is the final version of the paper which will be published in Studia Mathematica

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