Let $n \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be open, let $m \in \mathbb{R}$, and let
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\begin{align*}
a: U \times \mathbb{R}^n \to \mathbb{C}
\end{align*}
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be a smooth scalar symbol of order $m$: for every compact set $L \subset U$ and every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$, there exists a constant $C_{\alpha,\beta,L}>0$ such that
for all $(x,\xi) \in L \times \mathbb{R}^n$, where $\langle \xi \rangle=(1+|\xi|^2)^{1/2}$. Assume also that $a$ is elliptic of order $m$: for every compact set $L \subset U$, there exist constants $c_L>0$ and $R_L>0$ such that
for all $x \in L$ and all $\xi \in \mathbb{R}^n$ with $|\xi| \ge R_L$.
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Then for every compact set $K \subset U$, there exist an [open set](/page/Open%20Set) $V \subset \mathbb{R}^n$, a number $R>0$, and a constant $c>0$ such that $K \subset V \Subset U$ and
for every $x \in \overline V$ and every $\xi \in \mathbb{R}^n$ with $|\xi| \ge R$. In particular, $a(x,\xi) \neq 0$ for every $x \in V$ and every $\xi \in \mathbb{R}^n$ with $|\xi| \ge R$.
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For any such chosen $V$ and $R$, let $\chi \in C^\infty(\mathbb{R}^n)$ satisfy $\chi(\xi)=0$ for $|\xi| \le R$ and $\chi(\xi)=1$ for $|\xi| \ge 2R$. Define
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\begin{align*}
b: V \times \mathbb{R}^n \to \mathbb{C}
\end{align*}
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by setting $b(x,\xi)=0$ when $|\xi| \le R$ and $b(x,\xi)=\chi(\xi)a(x,\xi)^{-1}$ when $|\xi|>R$. Then
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\begin{align*}
b \in S^{-m}(V \times \mathbb{R}^n).
\end{align*}
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Thus the cutoff reciprocal of $a$ is a symbol of order $-m$ on a neighbourhood of $K$ compactly contained in $U$.