Translation-invariant continuous operators on S as convolution

Suppose $\Lambda : \mathscr {S}_{n} \rightarrow C\left(\mathbb {R}^{n}\right)$ is continuous, linear, and $\tau_{x} \Lambda = \Lambda \tau_{x}$ for every $x \in \mathbb {R}^{n}$. Does it follow that there exists $u \in \mathscr {S}_{n}^{\prime }$ such that

for every $\phi \in \mathscr {S}_{n}$?