Let $n \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be open, and write $\langle \xi \rangle := (1 + |\xi|^2)^{1/2}$ for $\xi \in \mathbb{R}^n$. For $s \in \mathbb{R}$, let $S^s(U \times \mathbb{R}^n)$ denote the standard symbol class $S^s_{1,0}$ of smooth functions $c: U \times \mathbb{R}^n \to \mathbb{C}$ such that, for every compact set $K \subset U$ and every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$, there is a constant $C_{K,\alpha,\beta} > 0$ satisfying

\begin{align*} |\partial_x^\alpha \partial_\xi^\beta c(x,\xi)| \leq C_{K,\alpha,\beta}\langle \xi \rangle^{s-|\beta|} \end{align*}

for all $(x,\xi) \in K \times \mathbb{R}^n$.

Let $m,m' \in \mathbb{R}$, let $a \in S^m(U \times \mathbb{R}^n)$, and let $b \in S^{m'}(U \times \mathbb{R}^n)$. Then, for every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$,

\begin{align*} \partial_x^\alpha \partial_\xi^\beta a \in S^{m-|\beta|}(U \times \mathbb{R}^n). \end{align*}

Moreover,

\begin{align*} ab \in S^{m+m'}(U \times \mathbb{R}^n). \end{align*}

If $\psi \in C^\infty(U)$, then the function $(x,\xi) \mapsto \psi(x)a(x,\xi)$ belongs to $S^m(U \times \mathbb{R}^n)$. If $\chi \in C^\infty(\mathbb{R}^n)$ satisfies the order-zero symbol estimates

\begin{align*} |\partial_\xi^\gamma \chi(\xi)| \leq C_\gamma \langle \xi \rangle^{-|\gamma|} \end{align*}

for every multi-index $\gamma \in \mathbb{N}_0^n$, some constant $C_\gamma > 0$, and all $\xi \in \mathbb{R}^n$, then the function $(x,\xi) \mapsto \chi(\xi)a(x,\xi)$ belongs to $S^m(U \times \mathbb{R}^n)$.