Hardy spaces of general Dirichlet series - a survey

Abstract

The main purpose of this article is to survey on some key elements of a recent ${\mathcal{H}}_p$-theory of general Dirichlet series $\sum a_n{e}^{-{\lambda }_{n}s}$, which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet series $\sum a_n{n}^{-s}$. In view of an ingenious identification of Bohr, the ${\mathcal{H}}_p$-theory of ordinary Dirichlet series can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus ${\mathbb{T}}^{\infty }$. Extending these ideas, the ${\mathcal{H}}_p$-theory of $\lambda$-Dirichlet series is build as a sub-theory of Fourier analysis on what we call $\lambda$-Dirichlet groups. A number of problems is added.