Hypercontractivity and strips of convergence in Hardy spaces of general Dirichlet series

Abstract

For a general Dirichlet series $\sum a_n{e}^{-{\lambda }_{n}s}$ with frequency $\lambda =({\lambda }_n{)}_n$, we study how horizontal translation (i.e. convolution with a Poisson kernel) improves its integrability properties. We characterize hypercontractive frequencies in terms of their additive structure answering some questions posed by Bayart. We also provide sharp bounds for the strips $S_p(\lambda )$ that encode the minimum translation necessary for series in the Hardy space ${\mathcal{H}}_p(\lambda )$ to have absolutely convergent coefficients.