Riesz means in Hardy spaces on Dirichlet groups

Abstract

Given a frequency $\lambda =({\lambda }_n)$, we study when almost all vertical limits of a ${\mathcal{H}}_1$-Dirichlet series $\sum a_n{e}^{-{\lambda }_{n}s}$ 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.