Hausdorff-Young type inequalities for vector-valued Dirichlet series

Abstract

We study Hausdorff-Young type inequalities for vector-valued Dirichlet series which allow to compare the norm of a Dirichlet series in the Hardy space ${\mathcal{H}}_p(X)$ with the $q$-norm of its coefficients. In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfy the restrictive notion of Fourier type/cotype. We show that variants of these inequalities hold for the much broader range of spaces enjoying type/cotype. We also consider Hausdorff-Young type inequalities for functions defined on the infinite torus ${\mathbb{T}}^{\infty }$ or the boolean cube $\{-1,1{\}}^{\infty }$.

The main contribution of the resubmitted version is that we have been able to show the equivalence between type/cotype and its polynomial counterpart. Therefore, the inequalities for vector-valued Dirichlet series that were obtained in the previous version actually hold assuming only type/cotype