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 Lp(Rd, | · | βp) into Lq(Rd, | · | -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 Td and Zd by a transference argument. We prove sharpness of our results by providing elementary examples on ℓp-spaces. Moreover, connections with Rademacher (co)type are discussed as well.
- Banach space
- Fourier type
- Limiting interpolation
- Vector-valued Fourier transform