Complex interpolation and geometry of Banach spaces (Q1066439)

A Banach space X is of G Fourier-type p, \(1\leq p\leq 2\), if the Fourier transform over G, F satisfies F:\(L^ p(G,X)\to L^{p'}(\hat G,X)\). Here \(f\in L^ p(G,X)\) iff f is X-valued, G a locally compact abelian group, \(\hat G\) the dual group, and \(1/p+1/p'=1\). This is a very useful concept, originally defined by Peetre, whose importance was overlooked for a number of years, until pointed out and applied by Milman. It enables one to prove complex interpolation results given the usually easier real interpolation results. The paper also contains applications to the geometry of Banach spaces, and shows once again how imaginative use of interpolation techniques reduces results in various areas of analysis to their bare essentials.