Let $U \subset \mathbb{R}^n$ be open, let $m \in \mathbb{N}$, and use Kohn-Nirenberg quantization on $U$. For $\mu \in \mathbb{R}$, let $S^\mu(U)$ denote the scalar local symbol class consisting of all $a \in C^\infty(U \times \mathbb{R}^n)$ 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 $C_{\alpha,\beta,K}>0$ with

\begin{align*} |\partial_x^\alpha\partial_\xi^\beta a(x,\xi)| \leq C_{\alpha,\beta,K}(1+|\xi|)^{\mu-|\beta|} \end{align*}

for every $(x,\xi) \in K \times \mathbb{R}^n$, and let $S^{-\infty}(U):=\bigcap_{N\in\mathbb{N}}S^{-N}(U)$. Let $\operatorname{Op}(a):\mathcal{D}'(U)\to\mathcal{D}'(U)$ denote the Kohn-Nirenberg pseudodifferential operator with scalar symbol $a$, and let $a\# b$ denote the Kohn-Nirenberg composition symbol of $\operatorname{Op}(a)\operatorname{Op}(b)$ whenever this composition is defined modulo smoothing operators.

Let

\begin{align*} L: \mathcal{D}'(U) \to \mathcal{D}'(U) \end{align*}

be a scalar differential operator of order $m$ with coefficients in $C^\infty(U)$. Let $p \in S^m(U)$ denote the full Kohn-Nirenberg symbol of $L$, and let $p_m:U\times(\mathbb{R}^n\setminus\{0\})\to\mathbb{C}$ denote its principal homogeneous symbol of degree $m$ in $\xi$. Assume that $p_m$ is elliptic on $U$ in the following compact-local sense: for every compact set $K \subset U$, there exist constants $C_K > 0$ and $R_K > 0$ such that

\begin{align*} |p_m(x,\xi)| \geq C_K |\xi|^m \end{align*}

for every $x \in K$ and every $\xi \in \mathbb{R}^n$ with $|\xi| \geq R_K$.

Then there exists a properly supported pseudodifferential operator

\begin{align*} Q: \mathcal{D}'(U) \to \mathcal{D}'(U) \end{align*}

of order $-m$ such that both remainders $QL - I$ and $LQ - I$ are smoothing operators on $U$, meaning that their Schwartz kernels are $C^\infty$ functions on $U\times U$.