Let $U \subset \mathbb{R}^n$ be open, let $m,m' \in \mathbb{R}$, and let $a \in S^m_{1,0}(U \times \mathbb{R}^n)$ and $b \in S^{m'}_{1,0}(U \times \mathbb{R}^n)$, where $S^s_{1,0}(U \times \mathbb{R}^n)$ denotes the standard Hörmander symbol class consisting of all $p \in C^\infty(U \times \mathbb{R}^n)$ 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 $C_{K,\alpha,\beta}>0$ with

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

for all $x \in K$ and $\xi \in \mathbb{R}^n$, with $\langle \xi\rangle=(1+|\xi|^2)^{1/2}$. For $u \in C_c^\infty(U)$, use the Kohn-Nirenberg convention

\begin{align*} \operatorname{Op}(p)u(x)=(2\pi)^{-n}\operatorname{Os}\!\int_U\int_{\mathbb{R}^n} e^{i(x-y)\cdot\xi}p(x,\xi)u(y)\,d\mathcal{L}^n(\xi)\,d\mathcal{L}^n(y). \end{align*}

Assume that $\operatorname{Op}(a)$ and $\operatorname{Op}(b)$ are properly supported on $U$. Then there exists a properly supported pseudodifferential operator $C$ on $U$ whose Kohn-Nirenberg symbol $a\# b \in S^{m+m'}_{1,0}(U \times \mathbb{R}^n)$ satisfies

\begin{align*} \operatorname{Op}(a)\operatorname{Op}(b)-C \end{align*}

is a smoothing operator on $U$, and $C-\operatorname{Op}(a\# b)$ is smoothing if $\operatorname{Op}(a\# b)$ is formed with any standard proper-support cut-off. Moreover, for every integer $N \geq 1$,

\begin{align*} a\# b-\sum_{|\alpha|<N}\frac{i^{-|\alpha|}}{\alpha!}\partial_\xi^\alpha a\,\partial_x^\alpha b \in S^{m+m'-N}_{1,0}(U \times \mathbb{R}^n). \end{align*}

Equivalently,

\begin{align*} a\# b \sim \sum_{\alpha \in \mathbb{N}_0^n}\frac{i^{-|\alpha|}}{\alpha!}\partial_\xi^\alpha a\,\partial_x^\alpha b \end{align*}

in the standard symbol asymptotic sense. In particular, modulo smoothing operators, the composition has Kohn-Nirenberg symbol $a\# b$.