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

Carando, Daniel; Marceca, Felipe; Sevilla-Peris, Pablo

doi:10.1090/tran/8147

Hausdorff–Young-type inequalities for vector-valued Dirichlet series
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by Daniel Carando, Felipe Marceca and Pablo Sevilla-Peris PDF

Trans. Amer. Math. Soc. 373 (2020), 5627-5652 Request permission

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 $\mathcal {H}_{p} (X)$ 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 a hypercontractive homogeneous polynomial type and cotype, a result of independent interest.

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