Let $n \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be open, and let $m,s \in \mathbb{R}$. Let $P \in \Psi^m_{1,0}(U)$ be a properly supported scalar Kohn--Nirenberg pseudodifferential operator acting on complex-valued distributions, and let $p \in S^m_{1,0}(U \times \mathbb{R}^n)$ be a full Kohn--Nirenberg symbol of $P$ in the given Euclidean coordinates. Assume that $P$ is elliptic: for every compact set $K \subset U$, there exist constants $c_K > 0$ and $R_K > 0$ such that

\begin{align*} |p(x,\xi)| \geq c_K \langle \xi \rangle^m \end{align*}

for every $x \in K$ and every $\xi \in \mathbb{R}^n$ with $|\xi| \geq R_K$, where $\langle \xi \rangle := (1 + |\xi|^2)^{1/2}$.

Let $Q \in \Psi^{-m}_{1,0}(U)$ be a properly supported parametrix for $P$ such that

\begin{align*} QP = I + R \end{align*}

on $\mathcal{D}'(U)$, with $R \in \Psi^{-\infty}(U)$. Let $K_Q$ and $K_R$ denote the Schwartz kernels of $Q$ and $R$, respectively. Let $\pi_2: U \times U \to U$ be the projection onto the second factor. Suppose $\chi,\psi,\eta \in C_c^\infty(U)$ satisfy

\begin{align*} \psi = 1 \quad \text{on a neighbourhood of } \pi_2(\operatorname{supp} K_Q \cap (\operatorname{supp}\chi \times U)) \end{align*}

and

\begin{align*} \eta = 1 \quad \text{on a neighbourhood of } \pi_2(\operatorname{supp} K_R \cap (\operatorname{supp}\chi \times U)). \end{align*}

Then, for every $N > 0$, there exists a constant $C > 0$ such that, for every $u \in \mathcal{D}'(U)$ with $\psi Pu \in H^{s-m}(\mathbb{R}^n)$ and $\eta u \in H^{s-N}(\mathbb{R}^n)$ after extension by zero from $U$ to $\mathbb{R}^n$, one has

\begin{align*} \|\chi u\|_{H^s(\mathbb{R}^n)} \leq C\left(\|\psi Pu\|_{H^{s-m}(\mathbb{R}^n)} + \|\eta u\|_{H^{s-N}(\mathbb{R}^n)}\right). \end{align*}

The constant $C$ may depend on $U$, $P$, $Q$, $R$, $\chi$, $\psi$, $\eta$, $s$, $N$, and finitely many relevant symbol and smoothing-kernel seminorms, but not on $u$.

Consequently, if $u \in \mathcal{D}'(U)$ and $Pu \in H^{s-m}_{\mathrm{loc}}(U)$, then $u \in H^s_{\mathrm{loc}}(U)$.