# Riesz means in Hardy spaces on Dirichlet groups

## TL;DR

This paper investigates the almost everywhere Riesz summability of Dirichlet series and Fourier series of $H_1$-functions on $ ext{Dirichlet}$ groups, introducing a new maximal operator and applying results to various function spaces.

## Contribution

It develops a weak-type Hardy-Littlewood maximal operator for $ ext{Dirichlet}$ groups and studies the convergence of Dirichlet series and Fourier series in this context.

## Key findings

- Almost all vertical limits of $ ext{Dirichlet}$ series are Riesz-summable almost everywhere.
- Established a weak-type $(1, ext{infinity})$ maximal operator for $ ext{Dirichlet}$ groups.
- Applications to $H_1$-functions on the infinite torus, Dirichlet series, and bounded holomorphic functions.

## Abstract

Given a frequency $\lambda=(\lambda_n)$, we study when almost all vertical limits of a $\mathcal{H}_1$-Dirichlet series $\sum a_n e^{-\lambda_ns}$ are Riesz-summable almost everywhere on the imaginary axis. Equivalently, this means to investigate almost everywhere convergence of Fourier series of $H_1$-functions on so-called $\lambda$-Dirichlet groups, and as our main technical tool we need to invent a weak-type $(1, \infty)$ Hardy-Littlewood maximal operator for such groups. Applications are given to $H_1$-functions on the infinite dimensional torus $\mathbb{T}^\infty$, ordinary Dirichlet series $\sum a_n n^{-s}$, as well as bounded and holomorphic functions on the open right half plane, which are uniformly almost periodic on every vertical line.