Suppose $X$ is a Banach space (we can suppose it's nice like UMD). Is $(1 - e^{itx})$ an $L_2$ multiplier for $X$-valued functions? If so, is there a uniform multiplier norm for all $t \in \mathbb{Z}$? If it helps, you can assume that we can multiply $(1 - e^{itx})$ by some bump function $\psi$, but $\psi$ must be the same for all $t$.
This is clearly true when $X$ is Hilbert space due to Plancherel, but I'm interested in a result for more general Banach spaces. The only theorem I know in a more general setting is Mikhlin multiplier theorem for scalar multipliers of UMD-valued functions, which would give that $(1 - e^{itx}) \psi(x)$ is a multiplier, but the norm would not be uniform as $t \to \infty$.
