Riesz projection and bounded mean oscillation for Dirichlet series

TL;DR

This paper investigates the boundedness of the Riesz projection on torus spaces, explores the duality of Hardy spaces of Dirichlet series, and analyzes partial sum operators, revealing new insights into their norms and relationships.

Contribution

It proves the norm of the Riesz projection is 1 only for p ≤ 2, solves a longstanding problem, and connects Dirichlet series in BMOA to Hardy space duality and partial sum operator norms.

Findings

Riesz projection norm is 1 only if p ≤ 2

H^p(T^∞) does not contain dual of H^1(T^∞) for p > 2

Partial sum norms grow like log log N for Dirichlet series

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 $\log\log N$.