Extensions of the vector-valued Hausdorff-Young inequalities

2019/04/16 by Dominguez, Oscar, Veraar, Mark · 1 citation
#42B05 #42B10 (Secondary) #46B20 (Primary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1904.07930

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(ℝ^d,|⋅|^βp) into L^q(ℝ^d,|⋅|^-γq) under certain condition on p,q,β and γ. 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 \mathbbT^d and ℤ^d by a transference argument. We prove sharpness of our results by providing elementary examples on ℓ^p-spaces. Moreover, connections with Rademacher (co)type are discussed as well.

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