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 H p ( X ) \mathcal {H}_{p} (X) with the q 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 ∞ \mathbb {T}^{\infty } or the boolean cube { − 1 , 1 } ∞ \{-1,1\}^{\infty } . 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.