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.