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Hypercontractivity and strips of convergence in Hardy spaces of general Dirichlet series (2401.09996v1)
Published 18 Jan 2024 in math.FA
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.