Classification and geometric aspects of vector valued Fourier transforms

It is shown that for any locally compact abelian group and 1 ≤ p ≤ 2, the Fourier type p norm with respect to of a bounded linear operator T between Banach spaces, denoted by ‖ T |ℱ p ‖, satisfies ‖ T |ℱ p ‖ ≤ ‖ T |ℱ p ‖, where is the direct product of ℤ 2 , ℤ 3 , ℤ 4 , … It is also shown that if is not of bounded order then C n p ‖ T |ℱ p ‖ ≤ ‖ T |ℱ p ‖, where is the circle group, n is a onnegative integer and C p = . From these inequalities, for any locally compact abelian group ‖ T |ℱ 2 ‖ ≤ ‖ T |ℱ 2 ‖, and moreover if is not of bounded order then ‖ T |ℱ 2 ‖ = ‖ T |ℱ 2 ‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)