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

## TL;DR

This paper extends Hausdorff-Young inequalities to vector-valued Dirichlet series, establishing conditions on Banach spaces for these inequalities to hold, and explores related inequalities on infinite-dimensional tori and boolean cubes.

## Contribution

It generalizes classical inequalities to vector-valued settings and broadens the class of Banach spaces for which these inequalities are valid.

## Key findings

- Inequalities hold for Banach spaces with type/cotype properties.
- Variants of inequalities are valid on infinite torus and boolean cube.
- Conditions for inequalities to mirror scalar case are characterized.

## 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 $\{-1,1\}^{\infty}$.