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Let $\mathbb{G}$ be a locally compact abelian group and let $1<p\leq 2$.$\mathbb{G}^{'}$ is the dual group of $\mathbb{G}$, and $p^{'}$ the conjugateexponent of $p$. An operator $T$ between Banach spaces $X$ and $Y$ is said tobe compatible with the Fourier transform $F^{\mathbb{G}}$ if$F^{\mathbb{G}}\otimes T: L_p(\mathbb{G})\otimes X\toL_{p^{'}}(\mathbb{G}^{'})\otimes Y $ admits a continuous extension$[F^{\mathbb{G}},T]:[L_p(\mathbb{G}),X]\to [L_{p^{'}}(\mathbb{G}^{'}),Y]$.$\mathcal{FT}_p^{\mathbb{G}}$ denotes the set of such $T$'s. We show that$\mathcal{FT}_p^{\mathbb{R}\times\mathbb{G}}=\mathcal{FT}_p^{\mathbb{Z}\times\mathbb{G}} =\mathcal{FT}_p^{\mathbb{Z}^n \times\mathbb{G}}$ for any $\mathbb{G}$and positive integer $n$. And if the factor group of $\mathbb{G}$ with respectto its component of the identity element is a direct sum of a torsion freegroup and a finite group with discrete topology then$\mathcal{FT}_p^{\mathbb{G}}=\mathcal{FT}_p^{\mathbb{Z}}$ .

On the vector-valued Fourier transform and compatibility of operators