$$\pmb {{\mathcal {H}}_{p}}$$-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, may 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.