Extensions of the vector-valued Hausdorff–Young inequalities

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