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Extensions of the vector-valued Hausdorff–Young inequalities

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Abstract

In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff–Young, Hardy–Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff–Young and Hardy–Littlewood inequalities and state that the Fourier transform is bounded from \(L^p({\mathbb {R}}^d,|\cdot |^{\beta p})\) into \(L^q({\mathbb {R}}^d,|\cdot |^{-\gamma q})\) under certain condition on \(p,q,\beta \) and \(\gamma \). Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and related geometric properties. Similar results are derived for \({\mathbb {T}}^d\) and \({\mathbb {Z}}^d\) by a transference argument. We prove sharpness of our results by providing elementary examples on \(\ell ^p\)-spaces. Moreover, connections with Rademacher (co)type are discussed as well.

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  1. Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    Oscar Dominguez

  2. Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2628 CD, Delft, The Netherlands

    Mark Veraar

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Oscar Dominguez is supported in part by MTM2017-84058-P (AEI/FEDER, UE). Mark Veraar is supported by the VIDI subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO). Part of the work was done during the visit of the first author to the Isaac Newton Institute for Mathematical Sciences, Cambridge, EPSCR Grant no EP/K032208/1.

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Dominguez, O., Veraar, M. Extensions of the vector-valued Hausdorff–Young inequalities. Math. Z. 299, 373–425 (2021). https://doi.org/10.1007/s00209-020-02675-6

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