Hausdorff-Young-type inequalities for vector-valued Dirichlet series

Abstract

We study Hausdorff-Young-type inequalities for vector-valued Dirichlet series which allow us to compare the norm of a Dirichlet series in the Hardy space Hp(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 T ∞ or the boolean cube {-1, 1}∞. As a fundamental tool we show that type and cotype are equivalent to a hypercontractive homogeneous polynomial type and cotype, a result of independent interest.