Let $U \subset \mathbb{R}^n$ be open, let $V \subset U$ be open, and let $m,s \in \mathbb{R}$. Let $P \in \Psi^m(U)$ be a properly supported pseudodifferential operator that is elliptic at every covector $(x,\xi) \in V \times (\mathbb{R}^n \setminus \{0\})$. If $u \in \mathcal{D}'(U)$ and $Pu \in H^s_{\mathrm{loc}}(V)$, then $u \in H^{s+m}_{\mathrm{loc}}(V)$.