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

中文翻译：

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向量值狄利克雷级数的 Hausdorff-Young 型不等式

我们研究了向量值狄利克雷级数的 Hausdorff-Young 类型不等式，它允许比较 Hardy 空间 $\mathcal{H}_{p} (X)$ 中狄利克雷级数的范数和它的 $q$-范数。系数。为了获得完全类似于标量情况的不等式，Banach 空间必须满足傅立叶类型/共型的限制性概念。我们表明，这些不等式的变体适用于更广泛的享受类型/共型的空间。我们还考虑了定义在无限环面 $\mathbb{T}^{\infty}$ 或布尔立方体 $\{-1,1\}^{\infty}$ 上的函数的 Hausdorff-Young 类型不等式。