Let $U \subset \mathbb{R}^n$ be open, and let $m,s \in \mathbb{R}$. Let $P \in \Psi^m(U)$ be an elliptic properly supported pseudodifferential operator. Suppose that $Q \in \Psi^{-m}(U)$ is properly supported and that $R \in \Psi^{-\infty}(U)$ is a properly supported smoothing operator such that, as operators on $\mathcal{E}'(U)$,

\begin{align*} QP = I + R. \end{align*}

If $u \in \mathcal{E}'(U)$ and $Pu \in H^s_{\mathrm{comp}}(U)$, then

\begin{align*} u = QPu - Ru \end{align*}

in $\mathcal{D}'(U)$, and $u \in H^{s+m}_{\mathrm{loc}}(U)$.