Classification and geometric aspects of vector valued Fourier transforms

Abstract

It is shown that for any locally compact abelian group G and 1 <= p <= 2, the Fourier type p norm with respect to G of a bounded linear operator T between Banach spaces, denoted by parallel to T vertical bar FT(p)(G)parallel to, satisfies parallel to T vertical bar FT(p)(G)parallel to <= parallel to T vertical bar FT(p)(A)parallel to where A is the direct product of Z(2), Z(3), Z(4), ... It is also shown that if G is not of bounded order then C(p)(n) parallel to T vertical bar FT(p)(T)parallel to <= parallel to T vertical bar FT(p)(G)parallel to, where T is the circle group, n is a nonnegative integer and C(p) = inf(theta is an element of R)(Sigma(k is an element of Z)vertical bar sin theta/theta + k pi vertical bar p'). From these inequalities, for any locally compact abelian group G parallel to T vertical bar FT(2)(G)parallel to <= parallel to T vertical bar FT(2)(T)parallel to, and moreover if G is not of bounded order then parallel to T vertical bar FT(2)(G)parallel to <= parallel to T vertical bar FT(2)(T)parallel to. The Hilbertian property and B-convexity are discussed in the framework of Fourier type p norms. (C) 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.