Let $U \subset \mathbb{R}^n$ be an [open set](/page/Open%20Set), let $m \in \mathbb{R}$, and let $P \in \Psi^m(U)$ be an elliptic pseudodifferential operator, under the standard convention that operators in $\Psi^m(U)$ act locally on distributions. Here $\mathcal{D}'(U)$ denotes the space of distributions on $U$, $C^\infty(U)$ denotes the space of smooth real-valued functions on $U$, $C_c^\infty(U)$ denotes the space of compactly supported smooth real-valued functions on $U$, and for each $r \in \mathbb{R}$, $H^r(\mathbb{R}^n)$ denotes the $L^2$-[Sobolev space](/page/Sobolev%20Space) of order $r$ on $\mathbb{R}^n$. If $u \in \mathcal{D}'(U)$ and $Pu$ is represented by a function in $C^\infty(U)$, then $u$ is represented by a function in $C^\infty(U)$.