On the vector valued Hausdorff-Young inequality

This paper studies Banach space valued Hausdorff-Young inequalities. The largest part considers ways of changing the underlying group. In particular the possibility to deduce the inequality for open subgroups as well as for quotient groups arising from compact subgroups is secured. A large body of results concerns the classical groupsT ⁿ ,R ⁿ andZ _k . Notions of Fourier type are introduced and they are shown to be equivalent to properties expressed by finite groups alone.