such $\phi$ is denoted ${M_{\mathcal{CB}}A}(G)$. Note that $T_\phi$ is the restriction of $M_\phi$ to ${\mathcal{L}}G \subset B(L^2G)$.

In this paper we investigate the Bo[ż]{}ejko-Fendler result for $L^p$-multipliers. When $G$ is discrete and $\phi \in {M_{\mathcal{CB}}A}(G)$, there is no technical difficulty for defining the Fourier multiplier $T_\phi^p:L^p({\mathcal{L}}G) \to L^p({\mathcal{L}}G)$ and the Schur multiplier $M_\phi^p:{\mathcal{S}}_p(\ell^2 G) \to {\mathcal{S}}_p(\ell^2 G)$ (they are just the extension of $T_\phi$/ restriction of $M_\phi$). When $G$ is not discrete, the definition of the Schur multiplier $M_\phi^p:{\mathcal{S}}_p(L^2 G) \to {\mathcal{S}}_p(L^2 G)$ is known (see Subsection \[subsection=Schur\]), and we define in Subsection \[subsection=Fourier\] the $L^p$-Fourier multiplier $T_\phi^p:L^p({\mathcal{L}}G) \to L^p({\mathcal{L}}G)$ spatially [@Con], [@Hilsum].

We are interested in the following question.

\[Que-One\] Let $G$ be a locally compact group. Is it true that, $$\Vert T_\phi^p \Vert_{{\mathcal{CB}}(L^p({\mathcal{L}}G))} = \Vert M_\phi^p \Vert_{{\mathcal{CB}}({\mathcal{S}}_p(L^2G))}.$$ for all completely bounded Fourier multipliers $\phi$ of $G$?

There are several motivations to study this problem. One is that a positive answer would imply that for a discrete group $G$, the property ${\rm AP}^{Schur}_{p,cb}$ considered in [@LafSal], see also [@Laa], is an invariant of the group von Neumann algebra of $G$. Furthermore, one is often interested in strict estimates of the norm of the transfered Fourier multiplier, see for example 