Let $T:S(\mathbb{R}^d)\to S(\mathbb{R}^d)$, a continuous linear operator, where $S(\mathbb{R}^d)$ is the Schwartz space. There is a result that guarantees that the family $F=\{\delta_s\circ T\}_{s\in\mathbb{R}^d}$ of functionals is a family of distribution and an $^S$family. That is, for each $u\in S(\mathbb{R}^d)$, the map $s\mapsto (\delta_s\circ T)(u)$ is in $S(\mathbb{R}^d)$.
Now, let $\varphi\in S(\mathbb{R}^{d(n+1)})$. For each $(s_1,...,s_n)\in\mathbb{R}^d$ we have $\varphi(\,\,,s_1,...,s_n)\in S(\mathbb{R}^d)$. Thus, let us take $h:\mathbb{R}^{d(n+1)}\to\mathbb{C}$ given by $$h(s,s_1,...,s_n)=(\delta_s\circ T)(\varphi(\,\,,s_1,...,s_n))=(T\varphi(\,\,,s_1,...,s_n))(s).$$
It is true that $h\in S(\mathbb{R}^d)$?