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

CARANDO, DANIEL; MARCECA, FELIPE; SEVILLA-PERIS, PABLO

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Año: 2020 vol. 373 p. 5627 - 5652

0002-9947

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 $\{-1,1\}^{\infty}$. {As a fundamental tool we show that type and cotype are equivalent to hypercontractive homogeneous polynomial type and cotype, a result of independent interest.}