$\mathcal{H}_p$-theory of general Dirichlet series

Abstract

Inspired by results of Bayart on ordinary Dirichlet series $\sum a_n n^{-s}$, the main purpose of this article is to start an $\mathcal{H}_p$-theory of general Dirichlet series $\sum a_n e^{-\lambda_{n}s}$. Whereas the $\mathcal{H}_p$-theory of ordinary Dirichlet series, in view of an ingenious identification of Bohr, can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus $\mathbb{T}^\infty$, the $\mathcal{H}_p$-theory of general Dirichlet series is build as a sub-theory of Fourier analysis on certain compact abelian groups, including the Bohr compactification $\overline{\mathbb{R}}$ of the reals. Our approach allows to extend various important facts on Hardy spaces of ordinary Dirichlet series to the much wider setting of $\mathcal{H}_p$-spaces of general Dirichlet series.