[proofplan]
We prove the local assertion on an arbitrary compact set $K \Subset V$. Ellipticity on a slightly larger compact subset of $V$ gives a localized parametrix: after inserting cutoffs, $u$ is equal near $K$ to a pseudodifferential operator of order $-m$ applied to $Pu$, plus a smoothing remainder applied to $u$. The mapping property of pseudodifferential operators sends the first term from $H^s$ to $H^{s+m}$, while the smoothing remainder is smooth. Since $K \Subset V$ is arbitrary, this gives $u \in H^{s+m}_{\mathrm{loc}}(V)$.
[/proofplan]
[step:Localize around an arbitrary compact subset of the elliptic region]
Fix a compact set $K \Subset V$. Choose an [open set](/page/Open%20Set) $W \subset U$ and functions $\chi,\psi \in C_c^\infty(V)$ such that $K \subset W$, $\overline{W} \Subset V$, $\chi = 1$ on $W$, and $\psi = 1$ on a neighbourhood of $\operatorname{supp}\chi$.
Let $M_\chi:\mathcal{D}'(U)\to \mathcal{D}'(U)$ and $M_\psi:\mathcal{D}'(U)\to \mathcal{D}'(U)$ denote multiplication by $\chi$ and $\psi$, respectively. Since $\chi=1$ on $W$, it is enough to prove $M_\chi u \in H^{s+m}(U)$, because then $u|_W=(M_\chi u)|_W$ and the restriction of $u$ to a neighbourhood of $K$ belongs to $H^{s+m}$.
[/step]
[step:Insert a localized elliptic parametrix]
By the localized [elliptic parametrix theorem for pseudodifferential operators](/theorems/7696), applied on the compact set $\operatorname{supp}\chi \Subset V$ where $P$ is elliptic, there exist a properly supported operator $Q \in \Psi^{-m}(U)$ and a smoothing operator
\begin{align*}
R:\mathcal{D}'(U) \to C^\infty(U)
\end{align*}
such that the localized identity
\begin{align*}
M_\chi = M_\chi Q M_\psi P + R
\end{align*}
holds as an identity of continuous linear maps $\mathcal{D}'(U)\to \mathcal{D}'(U)$. Here we are citing a result not yet in the wiki: localized elliptic parametrix theorem for pseudodifferential operators.
Applying this identity to $u \in \mathcal{D}'(U)$ gives
\begin{align*}
M_\chi u = M_\chi Q M_\psi Pu + R u .
\end{align*}
[guided]
We redeclare the localized objects used in this step. The functions $\chi,\psi \in C_c^\infty(V)$ are chosen so that $\operatorname{supp}\chi \Subset V$ and $\psi = 1$ on a neighbourhood of $\operatorname{supp}\chi$. The maps $M_\chi:\mathcal{D}'(U)\to\mathcal{D}'(U)$ and $M_\psi:\mathcal{D}'(U)\to\mathcal{D}'(U)$ are multiplication by $\chi$ and $\psi$, respectively. The purpose of these cutoffs is to use ellipticity only where it is available: since $P$ is elliptic on $V$, all symbol inversions used to build the parametrix occur inside the elliptic region.
The localized elliptic parametrix theorem applies because $P \in \Psi^m(U)$ is properly supported and elliptic at every covector over $\operatorname{supp}\chi \Subset V$, while $\psi$ is identically $1$ near $\operatorname{supp}\chi$. It gives a properly supported operator $Q \in \Psi^{-m}(U)$ and a smoothing operator
\begin{align*}
R:\mathcal{D}'(U) \to C^\infty(U)
\end{align*}
for which
\begin{align*}
M_\chi = M_\chi Q M_\psi P + R
\end{align*}
as continuous linear maps from $\mathcal{D}'(U)$ to $\mathcal{D}'(U)$. We are citing a result not yet in the wiki: localized elliptic parametrix theorem for pseudodifferential operators.
Applying this operator identity to the given distribution $u \in \mathcal{D}'(U)$ gives
\begin{align*}
M_\chi u = M_\chi Q M_\psi Pu + R u .
\end{align*}
This identity is the mechanism of elliptic regularity: near $K$, the distribution $u$ is recovered from $Pu$ by an operator of order $-m$, up to a smooth error.
[/guided]
[/step]
[step:Use the Sobolev mapping property on the parametrix term]
Since $Pu \in H^s_{\mathrm{loc}}(V)$ and $\psi \in C_c^\infty(V)$, multiplication by $\psi$ gives
\begin{align*}
M_\psi Pu \in H^s(U).
\end{align*}
For any $r \in \mathbb{R}$, let $H^r_{\mathrm{comp}}(U)$ denote the space of elements of $H^r(U)$ whose distributional support is a compact subset of $U$. Because $\operatorname{supp}(M_\psi Pu) \subset \operatorname{supp}\psi \Subset U$, we have
\begin{align*}
M_\psi Pu \in H^s_{\mathrm{comp}}(U).
\end{align*}
By the Sobolev mapping property for pseudodifferential operators, the operator $Q \in \Psi^{-m}(U)$ extends continuously as
\begin{align*}
Q:H^s_{\mathrm{comp}}(U) \to H^{s+m}_{\mathrm{loc}}(U).
\end{align*}
Here we are citing a result not yet in the wiki: Sobolev mapping property for pseudodifferential operators. Therefore
\begin{align*}
Q M_\psi Pu \in H^{s+m}_{\mathrm{loc}}(U).
\end{align*}
Multiplication by $\chi \in C_c^\infty(U)$ preserves Sobolev regularity, so
\begin{align*}
M_\chi Q M_\psi Pu \in H^{s+m}(U).
\end{align*}
[/step]
[step:Absorb the smoothing remainder]
The operator $R$ is smoothing, so by the standard regularity property of smoothing operators,
\begin{align*}
R u \in C^\infty(U).
\end{align*}
Here we are citing a result not yet in the wiki: smoothing operators map distributions to smooth functions. Since smooth functions belong locally to every [Sobolev space](/page/Sobolev%20Space), in particular
\begin{align*}
R u \in H^{s+m}_{\mathrm{loc}}(U).
\end{align*}
Because $M_\chi u = M_\chi Q M_\psi Pu + R u$, and both terms on the right-hand side belong to $H^{s+m}_{\mathrm{loc}}(U)$, we obtain
\begin{align*}
M_\chi u \in H^{s+m}_{\mathrm{loc}}(U).
\end{align*}
Since $M_\chi u$ has compact support in $U$, this is equivalently $M_\chi u \in H^{s+m}(U)$.
[/step]
[step:Conclude local Sobolev regularity on the elliptic region]
On the open set $W$ chosen above, $\chi=1$, so
\begin{align*}
u|_W = (M_\chi u)|_W .
\end{align*}
Thus $u|_W \in H^{s+m}(W)$. Since $W$ is an open neighbourhood of $K$, $u$ has $H^{s+m}$ regularity on a neighbourhood of $K$. Since the compact set $K \Subset V$ was arbitrary, the definition of local Sobolev regularity gives
\begin{align*}
u \in H^{s+m}_{\mathrm{loc}}(V).
\end{align*}
This proves the theorem.
[/step]
