Variants of a theorem of Helson on general Dirichlet series

TL;DR

This paper extends Helson's theorem on the convergence of general Dirichlet series within Hardy spaces, covering broader frequency conditions and establishing maximal inequalities, with applications to their structural theory.

Contribution

It generalizes Helson's theorem to Hardy spaces () for all p, including non-reflexive cases, under wider frequency conditions, and introduces relevant maximal inequalities.

Findings

Extended Helson's theorem to () for all p

Established maximal inequalities for Hardy spaces of Dirichlet series

Applied results to the structure theory of these Hardy spaces

Abstract

A result of Helson on general Dirichlet series $\sum a_{n} e^{-\lambda_{n}s}$ states that, whenever $(a_{n})$ is $2$-summable and $\lambda=(\lambda_{n})$ satisfies a certain condition introduced by Bohr, then for almost all homomorphism $\omega \colon (\mathbb{R},+) \to \mathbb{T}$ the Dirichlet series $\sum a_{n} \omega(\lambda_{n})e^{-\lambda_{n}s}$ converges on the open right half plane $[Re>0]$. For ordinary Dirichlet series $\sum a_n n^{-s}$ Hedenmalm and Saksman related this result with the famous Carleson-Hunt theorem on pointwise convergence of Fourier series, and Bayart extended it within his theory of Hardy spaces $\mathcal{H}_p$ of such series. The aim here is to prove variants of Helson's theorem within our recent theory of Hardy spaces $\mathcal{H}_{p}(\lambda),\,1\le p \le \infty,$ of general Dirichlet series. To be more precise, in the reflexive case $1 < p < \infty$ we extend Helson's result to Dirichlet series in $\mathcal{H}_{p}(\lambda)$ without any further condition on the frequency $\lambda$, and in the non-reflexive case $p=1$ to the wider class of frequencies satisfying the so-called Landau condition (more general than Bohr's condition). In both cases we add relevant maximal inequalities. Finally, we give several applications to the structure theory of Hardy spaces of general Dirichlet series.