[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*}[/step]

[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: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.[/step]

[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: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]