Vector valued Fourier transforms and Fourier type

Since its invention harmonic analysis plays a prominent rôle in the investigation of real and complex functions. It is desirable to have the same or similar methods available also for vector valued functions. A direct transfer of classical theorems to the vector valued case is, in general, only possible if one considers functions with values in spaces with additional structural properties. This opens up a fascinating interaction between the geometry of the spaces and harmonic analysis. A central question is whether, given a certain Banach space X and a locally compact abelian group G, the X-valued Fourier transform on G still satisfies a Hausdorff-Young inequality. Banach spaces having this property (for a certain p ∈ (1, 2]) are said to be of Fourier type p with respect to G. We will outline the theory around this property with particular emphasis on the connections between the structure of the groups and spaces on the one hand and properties of the Fourier transform on the other hand.