Let $n \in \mathbb{N}$, let $\mathcal{L}^n$ denote [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}^n$, let $u \in \mathcal{D}'(\mathbb{R}^n)$, and let $(x_0,\xi_0) \in T^*\mathbb{R}^n \setminus 0$, so $x_0 \in \mathbb{R}^n$ and $\xi_0 \in \mathbb{R}^n \setminus \{0\}$. Here $\mathcal{D}'(\mathbb{R}^n)$ denotes distributions on $\mathbb{R}^n$, $\mathcal{E}'(\mathbb{R}^n)$ denotes compactly supported distributions on $\mathbb{R}^n$, and $I: \mathcal{D}'(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)$ denotes the identity operator. The [Fourier transform](/page/Fourier%20Transform) of $v \in \mathcal{E}'(\mathbb{R}^n)$ is the smooth function $\hat v: \mathbb{R}^n \to \mathbb{C}$ defined by the standard distributional extension of the symmetric normalization

\begin{align*} \hat f(\xi)=(2\pi)^{-n/2}\int_{\mathbb{R}^n} f(x)e^{-ix\cdot \xi}\,d\mathcal{L}^n(x) \end{align*}

for $f \in C_c^\infty(\mathbb{R}^n)$. For each $m \in \mathbb{R}$, $S^m(\mathbb{R}^n \times \mathbb{R}^n)$ denotes the standard Kohn-Nirenberg symbol class consisting of all $a \in C^\infty(\mathbb{R}^n \times \mathbb{R}^n)$ such that for every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$ there is a constant $C_{\alpha,\beta}>0$ with

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

for all $(x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n$. The class $S^{-\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ means $\bigcap_{m \in \mathbb{R}}S^m(\mathbb{R}^n \times \mathbb{R}^n)$. For each $m \in \mathbb{R}$, $\Psi^m(\mathbb{R}^n)$ denotes the class of properly supported pseudodifferential operators obtained from symbols in $S^m(\mathbb{R}^n \times \mathbb{R}^n)$ in this calculus, and $\Psi^{-\infty}(\mathbb{R}^n)$ denotes the smoothing operators, equivalently operators with smooth Schwartz kernel. The notation $\operatorname{Op}(a)$ denotes a properly supported quantization of a symbol $a \in S^m(\mathbb{R}^n \times \mathbb{R}^n)$, with the order $m$ determined by $a$, chosen so that on compact spatial supports it agrees modulo an operator in $\Psi^{-\infty}(\mathbb{R}^n)$ with the Kohn-Nirenberg oscillatory formula using the Fourier transform normalization above. The principal symbol of $\operatorname{Op}(a) \in \Psi^m(\mathbb{R}^n)$ is the equivalence class of $a$ modulo $S^{m-1}(\mathbb{R}^n \times \mathbb{R}^n)$, and $\operatorname{Op}(a)$ is elliptic at $(x_0,\xi_0)$ if some representative $a$ satisfies $|a(x,\xi)| \geq c(1+|\xi|)^m$ on a neighbourhood of $x_0$ and a conic neighbourhood of $\xi_0$ for all sufficiently large $|\xi|$, for some constant $c>0$. Operator microsupport and conic support are understood in the standard symbol-theoretic sense: outside the indicated closed conic set, the complete symbol is smoothing in the frequency variable. An operator is microlocally smoothing on $U \times \Gamma$, where $U \subset \mathbb{R}^n$ is open and $\Gamma \subset \mathbb{R}^n \setminus \{0\}$ is open and conic, if every properly supported order-zero operator with operator microsupport compactly contained in $U \times \Gamma$ composes with it to a smoothing operator. Assume the following standard facts of this properly supported calculus: properly supported operators extend continuously to distributions and preserve smooth functions; smoothing operators map distributions to smooth functions; multiplication by a function in $C_c^\infty(\mathbb{R}^n)$ is a properly supported order-zero pseudodifferential operator with principal symbol equal to that function; principal symbols multiply under composition; the elliptic microlocal parametrix theorem holds; microlocal smoothing on $U_0\times\Gamma_0$ implies rapid conic Fourier decay after multiplication by any $\chi\in C_c^\infty(U_0)$ in every open cone whose closure in $\mathbb{R}^n\setminus\{0\}$ is contained in $\Gamma_0$; and if $R$ is microlocally smoothing on $U_0\times\Gamma_0$, then every properly supported $B\in\Psi^0(\mathbb{R}^n)$ with operator microsupport compactly contained in a smaller $U\times\Gamma\Subset U_0\times\Gamma_0$ satisfies $BR\in\Psi^{-\infty}(\mathbb{R}^n)$. The following conditions are equivalent.

1. There exist $\chi \in C_c^\infty(\mathbb{R}^n)$ with $\chi(x_0) \neq 0$ and an open conic neighbourhood $\Gamma \subset \mathbb{R}^n \setminus \{0\}$ of $\xi_0$ such that the Fourier transform $\widehat{\chi u}$ of the compactly supported distribution $\chi u \in \mathcal{E}'(\mathbb{R}^n)$ is rapidly decreasing in $\Gamma$: for every $N \in \mathbb{N}$ there exists a constant $C_N > 0$ such that

\begin{align*} |\widehat{\chi u}(\xi)| \leq C_N(1+|\xi|)^{-N} \end{align*}

for all $\xi \in \Gamma$.

2. There exists a properly supported pseudodifferential operator $A: C_c^\infty(\mathbb{R}^n) \to C^\infty(\mathbb{R}^n)$ with $A \in \Psi^0(\mathbb{R}^n)$, extended by duality to $A: \mathcal{D}'(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)$, such that $A$ is elliptic at $(x_0,\xi_0)$ and $Au \in C^\infty(\mathbb{R}^n)$. 3. There exist an open neighbourhood $U \subset \mathbb{R}^n$ of $x_0$ and an open conic neighbourhood $\Gamma \subset \mathbb{R}^n \setminus \{0\}$ of $\xi_0$ such that, for every properly supported operator $B \in \Psi^0(\mathbb{R}^n)$ whose operator microsupport is compactly contained in $U \times \Gamma$, the distribution $Bu$ belongs to $C^\infty(\mathbb{R}^n)$.