[proofplan]
The proof is the standard local parametrix argument. Proper support lets the cutoffs $\psi$ and $\eta$ remove all portions of $Pu$ and $u$ that cannot interact with $\chi$ through the kernels of $Q$ and $R$. The parametrix identity $QP = I + R$ then gives $\chi u = \chi Q\psi Pu - \chi R\eta u$. The first term is controlled by the Sobolev mapping theorem for pseudodifferential operators, while the second is controlled by the smoothing estimate for $R$. Finally, compactly supported distributions have finite negative Sobolev order, so the estimate implies the qualitative local regularity assertion.
[/proofplan]
[step:Use proper support to localize the kernels to the chosen cutoffs]
We first record the cutoff consequences of proper support. Since $Q$ and $R$ are properly supported, the sets
\begin{align*}
E_Q := \pi_2(\operatorname{supp} K_Q \cap (\operatorname{supp}\chi \times U))
\end{align*}
and
\begin{align*}
E_R := \pi_2(\operatorname{supp} K_R \cap (\operatorname{supp}\chi \times U))
\end{align*}
are compact subsets of $U$. By hypothesis, $\psi = 1$ on a neighbourhood of $E_Q$ and $\eta = 1$ on a neighbourhood of $E_R$.
Let $v \in \mathcal{D}'(U)$. The Schwartz kernel of the operator
\begin{align*}
\chi Q(1-\psi): \mathcal{D}'(U) \to \mathcal{D}'(U)
\end{align*}
is the distribution on $U \times U$ given by
\begin{align*}
(x,y) \mapsto \chi(x)K_Q(x,y)(1-\psi(y)).
\end{align*}
If $(x,y) \in \operatorname{supp} K_Q$ and $x \in \operatorname{supp}\chi$, then $y \in E_Q$, so $1-\psi(y)=0$ on a neighbourhood of this possible support. Hence this kernel is zero, and therefore
\begin{align*}
\chi Qv = \chi Q\psi v.
\end{align*}
Applying the same argument to $R$ and $\eta$ gives
\begin{align*}
\chi Rv = \chi R\eta v.
\end{align*}
[guided]
The reason for introducing $\psi$ and $\eta$ is that a properly supported kernel only sees a compact set of input variables when the output variable is restricted to $\operatorname{supp}\chi$. Define
\begin{align*}
E_Q := \pi_2(\operatorname{supp} K_Q \cap (\operatorname{supp}\chi \times U)).
\end{align*}
Thus $E_Q$ is exactly the set of points $y \in U$ that can contribute to $(Qv)(x)$ for some $x \in \operatorname{supp}\chi$. Proper support of $Q$ implies that this projected set is compact. Since $\psi = 1$ on a neighbourhood of $E_Q$, multiplication by $1-\psi$ removes only input values that cannot affect $\chi Qv$.
Formally, the kernel of $\chi Q(1-\psi)$ is
\begin{align*}
(x,y) \mapsto \chi(x)K_Q(x,y)(1-\psi(y)).
\end{align*}
For this kernel to be nonzero, one would need simultaneously $x \in \operatorname{supp}\chi$, $(x,y) \in \operatorname{supp}K_Q$, and $1-\psi(y) \neq 0$. The first two conditions imply $y \in E_Q$, while the last condition is impossible on the neighbourhood of $E_Q$ where $\psi=1$. Hence the kernel vanishes as a distribution, and therefore
\begin{align*}
\chi Qv = \chi Q\psi v.
\end{align*}
The same support argument applies to
\begin{align*}
E_R := \pi_2(\operatorname{supp} K_R \cap (\operatorname{supp}\chi \times U)).
\end{align*}
Since $\eta=1$ near $E_R$, the kernel of $\chi R(1-\eta)$ vanishes, and therefore
\begin{align*}
\chi Rv = \chi R\eta v.
\end{align*}
[/guided]
[/step]
[step:Apply the parametrix identity with the correct sign]
The parametrix identity is
\begin{align*}
QP = I + R
\end{align*}
as an identity on $\mathcal{D}'(U)$. Applying this identity to $u \in \mathcal{D}'(U)$ gives
\begin{align*}
QPu = u + Ru.
\end{align*}
Multiplying by $\chi$ and rearranging gives
\begin{align*}
\chi u = \chi QPu - \chi Ru.
\end{align*}
Using the localization identities from the previous step with $v = Pu$ and $v = u$, respectively, we obtain
\begin{align*}
\chi u = \chi Q\psi Pu - \chi R\eta u.
\end{align*}
Here $\chi u$, $\psi Pu$, and $\eta u$ are compactly supported distributions on $U$, and hence are identified with compactly supported distributions on $\mathbb{R}^n$ by extension by zero.
[/step]
[step:Estimate the parametrix term by Sobolev boundedness]
Define
\begin{align*}
A: C_c^\infty(\mathbb{R}^n) &\to C^\infty(\mathbb{R}^n)
\end{align*}
to be the localized operator obtained from $\chi Q\psi$ after extending compactly supported inputs and outputs by zero between $U$ and $\mathbb{R}^n$. Since $Q \in \Psi^{-m}_{1,0}(U)$ and multiplication by $\chi$ and $\psi$ is a pseudodifferential operation of order $0$, the localized operator $A$ is a compactly supported pseudodifferential operator of order $-m$ on $\mathbb{R}^n$.
By the Sobolev mapping theorem for pseudodifferential operators, since $A$ has order $-m$, there exists a constant $C_A > 0$, depending only on finitely many relevant seminorms of the localized symbol of $A$ and on $s,m,\chi,\psi,Q$, such that
\begin{align*}
\|\chi Q\psi Pu\|_{H^s(\mathbb{R}^n)} \leq C_A \|\psi Pu\|_{H^{s-m}(\mathbb{R}^n)}.
\end{align*}
This applies because $\psi Pu \in H^{s-m}(\mathbb{R}^n)$ by hypothesis.
[/step]
[step:Estimate the smoothing remainder from an arbitrarily low Sobolev norm]
Define
\begin{align*}
B: C_c^\infty(\mathbb{R}^n) &\to C^\infty(\mathbb{R}^n)
\end{align*}
to be the localized operator obtained from $\chi R\eta$ after extension by zero. Since $R \in \Psi^{-\infty}(U)$, the operator $B$ is smoothing with compactly supported Schwartz kernel on $\mathbb{R}^n \times \mathbb{R}^n$.
By the Sobolev smoothing estimate for operators with smooth compactly supported kernels, for the chosen $N > 0$ there exists a constant $C_B > 0$, depending on $s$, $N$, $\chi$, $\eta$, and finitely many seminorms of the localized smoothing kernel of $B$, such that
\begin{align*}
\|\chi R\eta u\|_{H^s(\mathbb{R}^n)} \leq C_B \|\eta u\|_{H^{s-N}(\mathbb{R}^n)}.
\end{align*}
This applies because $\eta u \in H^{s-N}(\mathbb{R}^n)$ by hypothesis.
[guided]
The remainder term is easier than the parametrix term because $R$ is infinitely smoothing. After multiplying on the left by $\chi$ and on the right by $\eta$, its Schwartz kernel becomes a smooth compactly supported function on $\mathbb{R}^n \times \mathbb{R}^n$. Thus the localized operator
\begin{align*}
B: C_c^\infty(\mathbb{R}^n) \to C^\infty(\mathbb{R}^n)
\end{align*}
defined by $B = \chi R\eta$ has a smooth compactly supported kernel.
A smoothing operator with compactly supported kernel is continuous between any two Sobolev spaces. In the present estimate, the input space is $H^{s-N}(\mathbb{R}^n)$ and the output space is $H^s(\mathbb{R}^n)$. Therefore, for the fixed number $N>0$, there is a constant $C_B>0$ such that
\begin{align*}
\|B(\eta u)\|_{H^s(\mathbb{R}^n)} \leq C_B\|\eta u\|_{H^{s-N}(\mathbb{R}^n)}.
\end{align*}
Since $B(\eta u)=\chi R\eta u$, this is exactly
\begin{align*}
\|\chi R\eta u\|_{H^s(\mathbb{R}^n)} \leq C_B\|\eta u\|_{H^{s-N}(\mathbb{R}^n)}.
\end{align*}
The point of allowing the loss parameter $N$ is that the smoothing term can absorb an arbitrarily rough compactly supported distribution. Later, when proving the qualitative local regularity statement, we choose $N$ large enough so that the compactly supported distribution $\eta u$ lies in $H^{s-N}(\mathbb{R}^n)$.
[/guided]
[/step]
[step:Combine the two estimates to obtain the local a priori bound]
From the identity
\begin{align*}
\chi u = \chi Q\psi Pu - \chi R\eta u
\end{align*}
and the triangle inequality in $H^s(\mathbb{R}^n)$, we get
\begin{align*}
\|\chi u\|_{H^s(\mathbb{R}^n)} \leq \|\chi Q\psi Pu\|_{H^s(\mathbb{R}^n)} + \|\chi R\eta u\|_{H^s(\mathbb{R}^n)}.
\end{align*}
Substituting the two estimates above gives
\begin{align*}
\|\chi u\|_{H^s(\mathbb{R}^n)} \leq C_A\|\psi Pu\|_{H^{s-m}(\mathbb{R}^n)} + C_B\|\eta u\|_{H^{s-N}(\mathbb{R}^n)}.
\end{align*}
With
\begin{align*}
C := \max\{C_A,C_B\}
\end{align*}
we obtain
\begin{align*}
\|\chi u\|_{H^s(\mathbb{R}^n)} \leq C\left(\|\psi Pu\|_{H^{s-m}(\mathbb{R}^n)} + \|\eta u\|_{H^{s-N}(\mathbb{R}^n)}\right).
\end{align*}
This proves the stated estimate.
[/step]
[step:Derive local Sobolev regularity from the estimate]
Assume now that $u \in \mathcal{D}'(U)$ and $Pu \in H^{s-m}_{\mathrm{loc}}(U)$. Let $\chi \in C_c^\infty(U)$ be arbitrary. By proper support, choose $\psi,\eta \in C_c^\infty(U)$ satisfying the kernel projection hypotheses for this $\chi$. Since $Pu \in H^{s-m}_{\mathrm{loc}}(U)$, the compactly supported distribution $\psi Pu$, extended by zero to $\mathbb{R}^n$, belongs to $H^{s-m}(\mathbb{R}^n)$.
Since $\eta u$ is a compactly supported distribution on $\mathbb{R}^n$, the finite-order Sobolev embedding property for compactly supported distributions gives a number $M \in \mathbb{R}$ such that
\begin{align*}
\eta u \in H^{-M}(\mathbb{R}^n).
\end{align*}
Choose $N>0$ so large that
\begin{align*}
s-N \leq -M.
\end{align*}
By the monotonicity of Sobolev spaces in the order parameter, this implies
\begin{align*}
\eta u \in H^{s-N}(\mathbb{R}^n).
\end{align*}
The estimate already proved therefore yields
\begin{align*}
\|\chi u\|_{H^s(\mathbb{R}^n)} < \infty.
\end{align*}
Thus $\chi u \in H^s(\mathbb{R}^n)$. Since $\chi \in C_c^\infty(U)$ was arbitrary, this proves
\begin{align*}
u \in H^s_{\mathrm{loc}}(U).
\end{align*}
[/step]
