On the Vector valued Fourier Transform And Compatibility of Operators

Let $\mathbb{G}$ be a locally compact abelian group and let $1<p\le 2$. ${\mathbb{G}}^{{}^{\mathrm{\prime }}}$ is the dual group of $\mathbb{G}$, and $p^{{}^{\mathrm{\prime }}}$ 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 $admitsacontinuousextension$[F^{$\mathbb{G}$},T]:[L_p($\mathbb{G}$),X]\to [L_{p^{‘}}($\mathbb{G}$^{’}),Y]$\mathrm{.}$$\mathcal{F}\mathcal{T}$_p^{$\mathbb{G}$}$denotesthesetofsuch$T${}^{\mathrm{\prime }}s\mathrm{.}Weshowthat$$\mathcal{F}\mathcal{T}$_p^{$\mathbb{R}$\times$\mathbb{G}$}=$\mathcal{F}\mathcal{T}$_p^{$\mathbb{Z}$\times\m athbb{G}} =$\mathcal{F}\mathcal{T}$_p^{$\mathbb{Z}$^n \times$\mathbb{G}$}$forany$$\mathbb{G}$$andpositiveinteger$n$\mathrm{.}Andifthefactorgroupof$$\mathbb{G}$$ 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 ${\mathcal{F}\mathcal{T}}_p^{\mathbb{G}}={\mathcal{F}\mathcal{T}}_p^{\mathbb{Z}}$ .