[proofplan] Apply the parametrix identity directly to the compactly supported distribution $u$ and solve for $u$. The term $QPu$ gains $m$ Sobolev derivatives because $Q$ has pseudodifferential order $-m$ and $Pu$ is compactly supported in $H^s$. The remainder term $Ru$ is smooth, hence belongs locally to every [Sobolev space](/page/Sobolev%20Space). Since $H^{s+m}_{\mathrm{loc}}(U)$ is a [vector space](/page/Vector%20Space), subtracting the two terms gives the claimed local regularity of $u$. [/proofplan]

[step:Apply the parametrix identity to the compactly supported distribution $u$]Let \begin{align*} v := Pu \in H^s_{\mathrm{comp}}(U). \end{align*} Since the identity $QP = I + R$ holds on $\mathcal{E}'(U)$ and $u \in \mathcal{E}'(U)$, applying it to $u$ gives \begin{align*} QPu = u + Ru \end{align*} in $\mathcal{D}'(U)$. Rearranging in the vector space $\mathcal{D}'(U)$ yields \begin{align*} u = QPu - Ru. \end{align*}[/step]

[guided]Define the distribution \begin{align*} v := Pu \in H^s_{\mathrm{comp}}(U). \end{align*} This definition is legitimate because $Pu$ is assumed to lie in $H^s_{\mathrm{comp}}(U)$. The parametrix relation is an operator identity on compactly supported distributions: \begin{align*} QP = I + R. \end{align*} Since $u \in \mathcal{E}'(U)$, we may substitute $u$ into this identity. This gives \begin{align*} QPu = (I + R)u. \end{align*} The identity operator satisfies $Iu = u$, so the right-hand side is \begin{align*} (I + R)u = u + Ru. \end{align*} Thus, in $\mathcal{D}'(U)$, \begin{align*} QPu = u + Ru. \end{align*} Because $\mathcal{D}'(U)$ is a vector space, we may subtract $Ru$ from both sides and obtain \begin{align*} u = QPu - Ru. \end{align*} The sign is important: the hypothesis is $QP = I + R$, so solving for $u$ gives $u = QPu - Ru$, not $u = QPu + Ru$.[/guided]

[step:Use the order of $Q$ to gain $m$ Sobolev derivatives] The Sobolev mapping theorem for properly supported pseudodifferential operators states that an operator in $\Psi^a(U)$ maps $H^r_{\mathrm{comp}}(U)$ continuously into $H^{r-a}_{\mathrm{loc}}(U)$ for all $r,a \in \mathbb{R}$; this prerequisite is not yet linked in the wiki. Apply this result with $a=-m$, $r=s$, and the properly supported operator $Q \in \Psi^{-m}(U)$. Since $v = Pu \in H^s_{\mathrm{comp}}(U)$, we obtain \begin{align*} QPu = Qv \in H^{s-(-m)}_{\mathrm{loc}}(U). \end{align*} Hence \begin{align*} QPu \in H^{s+m}_{\mathrm{loc}}(U). \end{align*} [/step]

[step:Use smoothing of $R$ to place $Ru$ in every local Sobolev space] The smoothing-operator regularization theorem states that a properly supported operator in $\Psi^{-\infty}(U)$ maps $\mathcal{E}'(U)$ into $C^\infty(U)$, and hence into $H^t_{\mathrm{loc}}(U)$ for every $t \in \mathbb{R}$; this prerequisite is not yet linked in the wiki. Since $R \in \Psi^{-\infty}(U)$ is properly supported and $u \in \mathcal{E}'(U)$, it follows that \begin{align*} Ru \in H^t_{\mathrm{loc}}(U) \end{align*} for every $t \in \mathbb{R}$. Taking $t=s+m$ gives \begin{align*} Ru \in H^{s+m}_{\mathrm{loc}}(U). \end{align*} [/step]

[step:Subtract the regular terms to conclude local Sobolev regularity of $u$] The space $H^{s+m}_{\mathrm{loc}}(U)$ is a vector space of distributions on $U$. From the previous two steps, \begin{align*} QPu \in H^{s+m}_{\mathrm{loc}}(U) \end{align*} and \begin{align*} Ru \in H^{s+m}_{\mathrm{loc}}(U). \end{align*} Therefore their difference belongs to the same space: \begin{align*} QPu - Ru \in H^{s+m}_{\mathrm{loc}}(U). \end{align*} Using the distributional identity \begin{align*} u = QPu - Ru, \end{align*} we conclude that \begin{align*} u \in H^{s+m}_{\mathrm{loc}}(U). \end{align*} This is exactly the asserted local Sobolev gain. [/step]