Let $n \in \mathbb{N}$, let $m \in \mathbb{R}$, let $U \subset \mathbb{R}^n$ be open, let $V \subset U$ be open, and let $K \subset V$ be compact. Let $P:C_c^\infty(U)\to C^\infty(U)$ be a properly supported classical pseudodifferential operator with $P \in \Psi^m(U)$, full Kohn-Nirenberg symbol $p \in S^m_{\mathrm{cl}}(U\times\mathbb{R}^n)$, and principal homogeneous symbol $p_m \in S^m_{\mathrm{cl}}(T^*U \setminus 0)$. Assume that $P$ is elliptic over $V$, meaning that

\begin{align*} p_m(x,\xi) \neq 0 \end{align*}

for every $x \in V$ and every $\xi \in \mathbb{R}^n \setminus \{0\}$.

Let $I:C_c^\infty(U)\to C^\infty(U)$ denote the identity operator, and let $S^{-\infty}(N\times\mathbb{R}^n)$ denote the space of symbols rapidly decreasing in $\xi$ with all derivatives locally uniformly for $x$ in the [open set](/page/Open%20Set) $N\subset U$.

Then there exist an open set $W \subset U$ and a properly supported operator $Q:C_c^\infty(U)\to C^\infty(U)$ with $Q \in \Psi^{-m}(U)$ such that

\begin{align*} K \subset W \subset \overline{W} \subset V, \end{align*}

and for every $\chi,\psi \in C_c^\infty(W)$ satisfying $\psi = 1$ on an open neighbourhood of $\operatorname{supp}\chi$, one has

\begin{align*} \chi(QP-I)\psi \in \Psi^{-\infty}(U) \end{align*}

and

\begin{align*} \chi(PQ-I)\psi \in \Psi^{-\infty}(U). \end{align*}

Equivalently, $Q$ is a two-sided microlocal parametrix for $P$ over $K$.