On the Vector valued Fourier Transform And Compatibility of Operators

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 conjugate exponent of $p$. An operator $T$ between Banach spaces $X$ and $Y$ is said to be compatible with the Fourier transform $F^{\mathbb{G}}$ if $F^{\mathbb{G}}\otimes T: L_p(\mathbb{G})\otimes X\to L_{p^{'}}(\mathbb{G}^{'})\otimes Y$ 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 $\mathcal{FT}_p^{\mathbb{R}\times\mathbb{G}}=\mathcal{FT}_p^{\mathbb{Z}\times\m athbb{G}} =\mathcal{FT}_p^{\mathbb{Z}^n \times\mathbb{G}}$ for any $\mathbb{G}$ and positive integer $n$. And if the factor group of $\mathbb{G}$ with respect to its component of the identity element is a direct sum of a torsion free group and a finite group with discrete topology then $\mathcal{FT}_p^{\mathbb{G}}=\mathcal{FT}_p^{\mathbb{Z}}$ .