Let $U \subset \mathbb{R}^n$ be open, let $m \in \mathbb{R}$, and let $P \in \Psi^m(U)$ be an elliptic pseudodifferential operator of order $m$ represented by a properly supported operator $P:\mathcal{D}'(U)\to\mathcal{D}'(U)$. For each $\varphi \in C_c^\infty(U)$, let $M_\varphi:\mathcal{D}'(U)\to\mathcal{E}'(U)$ denote multiplication by $\varphi$, followed by extension by zero to a compactly supported distribution on $\mathbb{R}^n$ when Sobolev norms are taken. For normed spaces $X$ and $Y$, let $\mathcal{L}(X,Y)$ denote the space of bounded linear maps from $X$ to $Y$, equipped with its operator norm. Let $K \Subset U$, and choose $\psi,\chi \in C_c^\infty(U)$ such that $\psi = 1$ on an open neighbourhood of $K$ and $\chi = 1$ on an open neighbourhood of $\operatorname{supp}\psi$. Assume the local properly supported pseudodifferential calculus provides a localized elliptic parametrix for this pair of cutoffs: there exist a properly supported operator $Q \in \Psi^{-m}(U)$ and a compactly supported smoothing operator $S \in \Psi^{-\infty}(U)$ such that, as an identity of operators $\mathcal{D}'(U)\to\mathcal{E}'(U)$ interpreted after multiplication by the indicated cutoffs and zero extension to $\mathbb{R}^n$, one has

\begin{align*} M_\psi = M_\psi Q M_\chi P + S M_\chi. \end{align*}

Assume moreover that the zero-extended operator $M_\psi Q$ is a compactly supported pseudodifferential operator of order $-m$ on $\mathbb{R}^n$, hence bounded $H^s(\mathbb{R}^n)\to H^{s+m}(\mathbb{R}^n)$ for every $s\in\mathbb{R}$, and that $S$ is compactly supported smoothing, hence bounded $H^{-N}(\mathbb{R}^n)\to H^{s+m}(\mathbb{R}^n)$ for every $s\in\mathbb{R}$ and $N>0$.

Then for every $s \in \mathbb{R}$ and every $N > 0$, there exists a constant $C = C(P,Q,S,\psi,\chi,s,N) > 0$ such that, for every $u \in \mathcal{D}'(U)$ for which $\chi Pu \in H^s(\mathbb{R}^n)$ and $\chi u \in H^{-N}(\mathbb{R}^n)$ after extension by zero outside $U$, one has

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