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Abstract

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\rightarrow L_{p'}({\mathbb G}')\otimes Y $ admits a continuous extension $[F^{{\mathbb G}},T]:[L_p({\mathbb G}),X]\rightarrow [L_{p'}({\mathbb G}'),Y]$. Let ${\mathcal FT}_p^{{\mathbb G}}$ denote the collection 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 $n$. Moreover, if the factor group of ${\mathbb G}$ by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then ${\mathcal FT}_p^{{\mathbb G}}={\mathcal FT}_p^{{\mathbb Z}}$.