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

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 ${\mathbb{T}}^{\mathrm{\infty }}$, 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 $\bar{\mathbb{R}}$ 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.