This thesis is devoted to a study of Fourier type and cotype of linear maps between Banach spaces or between operator spaces with respect to certain unimodular groups.
First, we define Fourier type and cotype of linear maps with respect to certain unimodular groups by measuring how well those maps behave in accordance with vector-valued Hausdorff-Young inequalities on the groups. The class of groups we are using here includes all locally compact abelian groups, compact groups and connected Lie groups. We develop a basic theory of Fourier type and cotype analogous to the Fourier type theory of Banach spaces with respect to locally compact abelian groups including the transference principle to open subgroups.
Secondly, we restrict our attention to the case of abelian groups. We check that our definitions and previously existing definitions are equivalent and that many results in the Banach space setting still hold for the operator space setting such as equivalence of Fourier types with respect to classical groups. Furthermore, we prove another transference principle and duality theorem for both cases.
Finally, we consider the Heisenberg group as an example of non-abelian and non-compact groups and give an equivalent definition of Fourier type and cotype using representation theory. We prove that Fourier type and cotype with respect to the Heisenberg group implies Fourier type with respect to classical abelian groups.