[proofplan] The proof is a localized parametrix argument. We insert cutoffs so that the elliptic parametrix for $P$ is applied only where $\chi = 1$, and all commutator or support-separated terms become smoothing operators. The pseudodifferential Sobolev mapping theorem controls the parametrix term by $\|\chi Pu\|_{H^s}$, while the smoothing remainder maps the prescribed low [Sobolev space](/page/Sobolev%20Space) $H^{-N}$ into the high Sobolev space $H^{s+m}$. Combining the two operator norm bounds gives the stated estimate. [/proofplan]

[step:Fix the local Sobolev convention and multiplication notation] For each $\varphi \in C_c^\infty(U)$, let $M_\varphi:\mathcal{D}'(U)\to \mathcal{E}'(U)$ denote multiplication by $\varphi$, so $M_\varphi u=\varphi u$. Since $\varphi u$ has compact support in $U$, we regard it as a compactly supported distribution on $\mathbb{R}^n$ by extension by zero. Thus $\|\varphi u\|_{H^r(\mathbb{R}^n)}$ is the standard Sobolev norm of this zero extension. For normed spaces $X$ and $Y$, let $\mathcal{L}(X,Y)$ denote the space of bounded linear maps from $X$ to $Y$, equipped with its operator norm. Choose an [open set](/page/Open%20Set) $V \subset U$ such that \begin{align*} \operatorname{supp}\psi \subset V \Subset U \end{align*} and such that $\chi = 1$ on $V$. This is possible because $\chi = 1$ on an open neighbourhood of the compact set $\operatorname{supp}\psi$. [/step]

[step:Use the localized cutoff parametrix supplied by the calculus]By the localized elliptic parametrix hypothesis in the theorem statement, there exist a properly supported operator $Q \in \Psi^{-m}(U)$ and a compactly supported smoothing operator $S \in \Psi^{-\infty}(U)$ such that \begin{align*} M_\psi = M_\psi Q M_\chi P + S M_\chi \end{align*} as an identity of operators $\mathcal{D}'(U)\to\mathcal{E}'(U)$, interpreted after multiplication by the indicated cutoffs and zero extension to $\mathbb{R}^n$. Applying this distributional identity to $u \in \mathcal{D}'(U)$ gives \begin{align*} \psi u = M_\psi Q(\chi Pu) + S(\chi u). \end{align*}[/step]

[guided]The proof uses the cutoff parametrix as an explicit hypothesis, rather than hiding the needed support bookkeeping inside the phrase "microlocally on $V$". The hypothesis gives a properly supported operator $Q \in \Psi^{-m}(U)$ and a compactly supported smoothing operator $S \in \Psi^{-\infty}(U)$ satisfying the exact localized operator identity \begin{align*} M_\psi = M_\psi Q M_\chi P + S M_\chi \end{align*} from $\mathcal{D}'(U)$ to $\mathcal{E}'(U)$, interpreted after the displayed cutoffs and zero extension to $\mathbb{R}^n$. This identity already includes the support-separated remainders and the cutoff commutator terms, and it records the essential point that every remainder depends on the distribution only through $M_\chi u=\chi u$. Now let $u \in \mathcal{D}'(U)$. Evaluating the operator identity at $u$ gives \begin{align*} M_\psi u = M_\psi Q M_\chi P u + S M_\chi u. \end{align*} By the definition of $M_\psi$ and $M_\chi$, this is exactly \begin{align*} \psi u = M_\psi Q(\chi Pu) + S(\chi u). \end{align*} This is the central decomposition: the first term is controlled by the Sobolev regularity of the localized differentiated distribution $\chi Pu$, and the second term is controlled by the low Sobolev norm of the localized distribution $\chi u$.[/guided]

[step:Estimate the parametrix term in $H^{s+m}$] By the zero-extension convention and the mapping hypothesis in the theorem statement, the operator $M_\psi Q$ is represented as a compactly supported pseudodifferential operator of order $-m$ on $\mathbb{R}^n$. Therefore it extends to a bounded [linear map](/page/Linear%20Map) \begin{align*} M_\psi Q:H^s(\mathbb{R}^n)\to H^{s+m}(\mathbb{R}^n). \end{align*} Define \begin{align*} C_1:=\|M_\psi Q\|_{\mathcal{L}(H^s(\mathbb{R}^n),H^{s+m}(\mathbb{R}^n))}. \end{align*} Since $\chi Pu \in H^s(\mathbb{R}^n)$ by hypothesis, we obtain \begin{align*} \|M_\psi Q(\chi Pu)\|_{H^{s+m}(\mathbb{R}^n)} \leq C_1\|\chi Pu\|_{H^s(\mathbb{R}^n)}. \end{align*} [/step]

[step:Estimate the smoothing remainder from the low Sobolev norm] By the smoothing mapping hypothesis in the theorem statement, the compactly supported smoothing operator $S$ gives a bounded linear map \begin{align*} S:H^{-N}(\mathbb{R}^n)\to H^{s+m}(\mathbb{R}^n). \end{align*} Define \begin{align*} C_2:=\|S\|_{\mathcal{L}(H^{-N}(\mathbb{R}^n),H^{s+m}(\mathbb{R}^n))}. \end{align*} Since $\chi u \in H^{-N}(\mathbb{R}^n)$ by hypothesis, we have \begin{align*} \|S(\chi u)\|_{H^{s+m}(\mathbb{R}^n)} \leq C_2\|\chi u\|_{H^{-N}(\mathbb{R}^n)}. \end{align*} [/step]

[step:Combine the two bounds to obtain the localized elliptic estimate] From the decomposition \begin{align*} \psi u = M_\psi Q(\chi Pu) + S(\chi u) \end{align*} and the triangle inequality in the [Hilbert space](/page/Hilbert%20Space) $H^{s+m}(\mathbb{R}^n)$, we get \begin{align*} \|\psi u\|_{H^{s+m}(\mathbb{R}^n)} \leq \|M_\psi Q(\chi Pu)\|_{H^{s+m}(\mathbb{R}^n)} + \|S(\chi u)\|_{H^{s+m}(\mathbb{R}^n)}. \end{align*} Using the two operator bounds above, \begin{align*} \|\psi u\|_{H^{s+m}(\mathbb{R}^n)} \leq C_1\|\chi Pu\|_{H^s(\mathbb{R}^n)} + C_2\|\chi u\|_{H^{-N}(\mathbb{R}^n)}. \end{align*} Let \begin{align*} C:=\max\{C_1,C_2\}. \end{align*} Then \begin{align*} \|\psi u\|_{H^{s+m}(\mathbb{R}^n)} \leq C\left(\|\chi Pu\|_{H^s(\mathbb{R}^n)} + \|\chi u\|_{H^{-N}(\mathbb{R}^n)}\right). \end{align*} The constant depends on $P,Q,S,\psi,\chi,s,N$ through the chosen local parametrix and the corresponding operator norms. Since $\psi=1$ on an open neighbourhood of $K$, this estimate controls $u$ microlocally on the prescribed compact set $K$ through the localized distribution $\psi u$. This is the desired a priori estimate. [/step]