[proofplan]
We localize the output by a fixed cutoff $\chi$ and use proper support to identify a compact set of input points that can influence $\operatorname{supp}\chi$ through the Schwartz kernel of $A$. We then choose $\psi$ equal to $1$ on this compact input set, so that $\chi Au=\chi A(\psi u)$ distributionally; this eliminates the off-support remainder instead of merely estimating it. Finally, the doubly localized operator $\chi A\psi$ is represented by a compactly supported global pseudodifferential operator on $\mathbb{R}^n$, and the global Sobolev [boundedness theorem](/theorems/181) gives the desired seminorm estimate. Since these seminorm estimates are exactly the defining continuity estimates for the local Sobolev topology, the operator extends continuously.
[/proofplan]
[step:Choose the input cutoff using proper support]
Fix $\chi \in C_c^\infty(U)$ and define the compact output set $K_\chi := \operatorname{supp}\chi \subset U$. Let $K_A \subset U \times U$ denote the support of the Schwartz kernel of $A$. Since $A$ is properly supported, the coordinate projection
\begin{align*}
\pi_2: U \times U \to U
\end{align*}
maps $K_A \cap (K_\chi \times U)$ to a compact subset of $U$. Define
\begin{align*}
K_{\chi,A}:=\pi_2\bigl(K_A \cap (K_\chi \times U)\bigr).
\end{align*}
Choose $\psi \in C_c^\infty(U)$ such that $\psi=1$ on an open neighbourhood of $K_{\chi,A}$. This is possible because $K_{\chi,A}$ is compact and contained in the [open set](/page/Open%20Set) $U$.
[guided]
The role of proper support is to make the set of relevant input points compact. We have fixed the output cutoff $\chi$, so only values of $Au$ on $K_\chi=\operatorname{supp}\chi$ matter. The Schwartz kernel of $A$ has support $K_A\subset U\times U$, and a pair $(x,y)\in K_A$ means that the input near $y$ may influence the output near $x$.
The proper support hypothesis says precisely that the two coordinate projections are proper on $K_A$. Therefore the second projection
\begin{align*}
\pi_2:U\times U\to U
\end{align*}
sends the closed subset $K_A\cap(K_\chi\times U)$ to a compact subset of $U$. We denote this compact input set by
\begin{align*}
K_{\chi,A}:=\pi_2\bigl(K_A\cap(K_\chi\times U)\bigr).
\end{align*}
Thus every input point that can affect the localized output $\chi Au$ lies in $K_{\chi,A}$. Since $K_{\chi,A}$ is compactly contained in $U$, the cutoff lemma for smooth functions gives a function $\psi\in C_c^\infty(U)$ satisfying $\psi=1$ on an open neighbourhood of $K_{\chi,A}$. This is the cutoff that will appear in the right-hand side seminorm.
[/guided]
[/step]
[step:Eliminate the off-support term by the kernel support condition]
Let $M_{1-\psi}:C^\infty(U)\to C^\infty(U)$ be multiplication by $1-\psi$, and let $M_\chi:C^\infty(U)\to C_c^\infty(U)$ be multiplication by $\chi$. The Schwartz kernel of $M_\chi A M_{1-\psi}$ is
\begin{align*}
(x,y)\mapsto \chi(x)(1-\psi(y))K_A(x,y)
\end{align*}
in the standard distributional kernel notation. If $x\in \operatorname{supp}\chi$ and $(x,y)\in K_A$, then $y\in K_{\chi,A}$, hence $\psi(y)=1$. Therefore the support of this kernel is empty, so
\begin{align*}
M_\chi A M_{1-\psi}=0
\end{align*}
as an operator from distributions on $U$ to distributions on $U$. Consequently, for every $u\in H^s_{\mathrm{loc}}(U)$,
\begin{align*}
\chi Au=\chi A(\psi u)
\end{align*}
in $\mathcal{D}'(U)$.
[/step]
[step:Represent the localized operator as a global pseudodifferential operator]
Choose $\theta\in C_c^\infty(U)$ such that $\theta=1$ on an open neighbourhood of $\operatorname{supp}\chi\cup\operatorname{supp}\psi$. Let $E:C_c^\infty(U)\to C_c^\infty(\mathbb{R}^n)$ denote extension by zero, and let $R:C^\infty(\mathbb{R}^n)\to C^\infty(U)$ denote restriction to $U$. Define the global operator
\begin{align*}
B:=E M_\chi A M_\psi R
\end{align*}
initially on $C_c^\infty(\mathbb{R}^n)$. Since $M_\chi A M_\psi$ has compactly supported kernel contained in $\operatorname{supp}\chi\times\operatorname{supp}\psi$, the standard localization property of $\Psi^m_{1,0}$ gives
\begin{align*}
B\in \Psi^m_{1,0}(\mathbb{R}^n).
\end{align*}
Here we are invoking the local-to-global compact-kernel representation for pseudodifferential operators; citing a result not yet in the wiki: compactly supported local pseudodifferential kernels extend by zero to global pseudodifferential operators of the same order.
For every $u\in H^s_{\mathrm{loc}}(U)$, the distribution $\psi u$ extends by zero to an element of $H^s(\mathbb{R}^n)$ by the definition of $H^s_{\mathrm{loc}}(U)$, and
\begin{align*}
E(\chi Au)=B(E(\psi u))
\end{align*}
in $\mathcal{D}'(\mathbb{R}^n)$.
[/step]
[step:Apply the global Sobolev boundedness theorem]
By the global Sobolev boundedness theorem for pseudodifferential operators of type $(1,0)$, applied to the operator $B\in\Psi^m_{1,0}(\mathbb{R}^n)$, there exists a constant $C_B>0$ such that
\begin{align*}
\|Bv\|_{H^{s-m}(\mathbb{R}^n)}\le C_B\|v\|_{H^s(\mathbb{R}^n)}
\end{align*}
for every $v\in H^s(\mathbb{R}^n)$. Here we are invoking a standard result not yet resolved to a wiki theorem: global Sobolev boundedness for pseudodifferential operators.
Taking $v=E(\psi u)$ gives
\begin{align*}
\|\chi Au\|_{H^{s-m}(\mathbb{R}^n)}=\|B(E(\psi u))\|_{H^{s-m}(\mathbb{R}^n)}\le C_B\|\psi u\|_{H^s(\mathbb{R}^n)}.
\end{align*}
Thus the required estimate holds with $C:=C_B$.
[/step]
[step:Conclude continuity on local Sobolev spaces]
The topology of $H^s_{\mathrm{loc}}(U)$ is generated by the seminorms
\begin{align*}
u\mapsto \|\rho u\|_{H^s(\mathbb{R}^n)}
\end{align*}
where $\rho\in C_c^\infty(U)$ and products are extended by zero. The estimate just proved says that each output seminorm
\begin{align*}
u\mapsto \|\chi Au\|_{H^{s-m}(\mathbb{R}^n)}
\end{align*}
is bounded by one input seminorm. Hence $A:H^s_{\mathrm{loc}}(U)\to H^{s-m}_{\mathrm{loc}}(U)$ is continuous and linear.
Since $C_c^\infty(U)$ is dense in $H^s_{\mathrm{loc}}(U)$ in the local Sobolev topology after cutoff localization, the continuous operator obtained above is the unique extension of the original action of $A$ on test functions. This proves the theorem.
[/step]
