On the vector-valued Fourier transform and compatibility of operators

Let G be a locally compact abelian group and let 1 < p < 2. G' is the dual group of G, and p' the conjugate exponent of p. An operator T between Banach spaces X and Y is said to be compatible with the Fourier transform F-G if F-G circle times T : L-p (G) circle times X -> L-p' (G') circle times Y admits a continuous extension [F-G, T] : [L-p (G), X] -> [L-p' (G(')), Y)]. Let FTpG denote the collection of such T's. We show that FTpR (x G) = FTpZ (x G) = FTpZx G for any G and positive integer n. Moreover, if the factor group of G by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then FTpG =: FTpZ.