On the Vector Valued Fourier Transform and Compatibility of Operators*

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 ⊗ T : Lp(G)⊗X → Lp′ (G ′ )⊗ Y admits a continuous extension [F , T ] : [Lp(G), X ] → [Lp′ (G ′ ), Y ]. We show that if G is topologically isomorphic with R×Z×F, where l and m are nonnegative integers and F is a compact group with finitely many components, then an operator is compatible with F G if and only if it is compatible with F . And the same statement holds if G has finitely many components, or is of the form R×[torsion-free group with discrete topology]×[compact group with finite components].