[proofplan] We construct a Fourier parametrix for $P(D)$ at high frequencies: the multiplier obtained by cutting off $1/P(\xi)$ away from the zero-frequency region has order $-m$, while the complementary low-frequency multiplier is smoothing. Applying this parametrix to $\chi u$ reduces the $H^{r+m}$ norm of $\chi u$ to the $H^r$ norm of $P(D)(\chi u)$ plus a smoothing remainder. The commutator identity $P(D)(\chi u)=\chi P(D)u + [P(D),\chi]u$ then localizes the estimate, because the commutator has order at most $m-1$ and is supported where $\chi_1=1$. Finally, compactly supported distributions start in some sufficiently negative [Sobolev space](/page/Sobolev%20Space), so the estimate can be iterated with nested cutoffs until the desired local order $s+m$ is reached. [/proofplan]

[step:Choose a high-frequency inverse for the elliptic multiplier] Since $P_m(\xi) \neq 0$ for every $\xi \in \mathbb{R}^n \setminus \{0\}$ and the unit sphere is compact, there exists $c_0>0$ such that \begin{align*} |P_m(\omega)| \geq c_0 \end{align*} for every $\omega \in \mathbb{R}^n$ with $|\omega|=1$. Since $P(\xi)=P_m(\xi)+O(|\xi|^{m-1})$ as $|\xi|\to\infty$, there exist constants $R_0 \geq 1$ and $c_1>0$ such that \begin{align*} |P(\xi)| \geq c_1 |\xi|^m \end{align*} whenever $|\xi| \geq R_0$. Choose a function $\psi \in C^\infty(\mathbb{R}^n)$ such that $0 \leq \psi \leq 1$, $\psi(\xi)=0$ for $|\xi|\leq R_0$, and $\psi(\xi)=1$ for $|\xi|\geq 2R_0$. Define the high-frequency inverse symbol \begin{align*} q: \mathbb{R}^n &\to \mathbb{C} \end{align*} by setting \begin{align*} q(\xi) = \frac{\psi(\xi)}{P(\xi)} \end{align*} for every $\xi \in \mathbb{R}^n$. This is a smooth function because $\psi$ vanishes on the region where the denominator might vanish. Moreover there is a constant $C_q>0$ such that \begin{align*} |q(\xi)| \leq C_q \langle \xi\rangle^{-m} \end{align*} for every $\xi \in \mathbb{R}^n$, where $\langle \xi\rangle = (1+|\xi|^2)^{1/2}$. Let $Q:\mathcal{S}'(\mathbb{R}^n)\to\mathcal{S}'(\mathbb{R}^n)$ be the Fourier multiplier with symbol $q$, and let $R:\mathcal{S}'(\mathbb{R}^n)\to\mathcal{S}'(\mathbb{R}^n)$ be the Fourier multiplier with symbol $1-\psi$. For every $a \in \mathbb{R}$, the bound on $q$ gives \begin{align*} \|Qv\|_{H^{a+m}(\mathbb{R}^n)} \leq C_q \|v\|_{H^a(\mathbb{R}^n)} \end{align*} for every $v \in H^a(\mathbb{R}^n)$. Also, since $1-\psi$ is smooth and compactly supported in frequency, for every $a,b \in \mathbb{R}$ there is a constant $C_{R,a,b}>0$ such that \begin{align*} \|Rv\|_{H^b(\mathbb{R}^n)} \leq C_{R,a,b}\|v\|_{H^a(\mathbb{R}^n)} \end{align*} for every $v \in H^a(\mathbb{R}^n)$. Finally, define $\psi(D):\mathcal{S}'(\mathbb{R}^n)\to\mathcal{S}'(\mathbb{R}^n)$ to be the Fourier multiplier with symbol $\psi$. In $\mathcal{S}'(\mathbb{R}^n)$, \begin{align*} QP(D)v = \psi(D)v \end{align*} and therefore \begin{align*} v = QP(D)v + Rv \end{align*} for every $v \in \mathcal{S}'(\mathbb{R}^n)$. [/step]

[step:Record the cutoff multiplier estimate used for localization]We shall use the following standard cutoff estimate, whose proof follows from the rapid decay of the [Fourier transform](/page/Fourier%20Transform) of a compactly supported smooth function and Peetre's inequality: for every $\varphi \in C_c^\infty(\mathbb{R}^n)$ and every $t \in \mathbb{R}$, multiplication by $\varphi$ defines a bounded [linear map](/page/Linear%20Map) \begin{align*} M_\varphi: H^t(\mathbb{R}^n) &\to H^t(\mathbb{R}^n) \end{align*} \begin{align*} v &\mapsto \varphi v. \end{align*} Thus there exists $C_{\varphi,t}>0$ such that \begin{align*} \|\varphi v\|_{H^t(\mathbb{R}^n)} \leq C_{\varphi,t}\|v\|_{H^t(\mathbb{R}^n)} \end{align*} for every $v \in H^t(\mathbb{R}^n)$.[/step]

[guided]Why is multiplication by a smooth cutoff harmless on Sobolev spaces of arbitrary real order? Let $\varphi \in C_c^\infty(\mathbb{R}^n)$ and $t \in \mathbb{R}$. The Fourier transform of $\varphi$ is a Schwartz function, so for every $N \in \mathbb{N}$ there is a constant $A_N>0$ such that \begin{align*} |\widehat{\varphi}(\eta)| \leq A_N \langle \eta\rangle^{-N} \end{align*} for every $\eta \in \mathbb{R}^n$. For $v \in \mathcal{S}(\mathbb{R}^n)$, the Fourier transform of $\varphi v$ is the convolution of $\widehat{\varphi}$ and $\widehat{v}$ up to the harmless normalization constant determined by the Fourier transform convention. Peetre's inequality gives a constant $C_t>0$ such that \begin{align*} \langle \xi\rangle^t \leq C_t \langle \xi-\eta\rangle^{|t|}\langle \eta\rangle^t \end{align*} for all $\xi,\eta \in \mathbb{R}^n$. Choosing $N>n+|t|+1$, the kernel $\eta \mapsto \langle \eta\rangle^{|t|}|\widehat{\varphi}(\eta)|$ belongs to $L^1(\mathbb{R}^n)$. [Young's convolution inequality](/theorems/463) then gives \begin{align*} \|\varphi v\|_{H^t(\mathbb{R}^n)} \leq C_{\varphi,t}\|v\|_{H^t(\mathbb{R}^n)}. \end{align*} Since $\mathcal{S}(\mathbb{R}^n)$ is dense in $H^t(\mathbb{R}^n)$ for finite $t$, the multiplication operator extends uniquely and continuously to all of $H^t(\mathbb{R}^n)$. This is the analytic reason cutoff functions can be inserted and removed in the local estimate without changing Sobolev order.[/guided]

[step:Apply the parametrix to the localized distribution] Let $r \in \mathbb{R}$ and let $\chi,\chi_1 \in C_c^\infty(\mathbb{R}^n)$ satisfy $\chi_1=1$ on an open neighbourhood of $\operatorname{supp}\chi$. Let $u \in \mathcal{S}'(\mathbb{R}^n)$ satisfy \begin{align*} \chi_1P(D)u \in H^r(\mathbb{R}^n) \end{align*} and \begin{align*} \chi_1u \in H^{r+m-1}(\mathbb{R}^n). \end{align*} Apply the identity from the previous step to $v=\chi u$. Since $\chi u = \chi\chi_1u$ and $H^{r+m-1}(\mathbb{R}^n) \subset \mathcal{S}'(\mathbb{R}^n)$, the expression is meaningful in $\mathcal{S}'(\mathbb{R}^n)$: \begin{align*} \chi u = QP(D)(\chi u) + R(\chi u). \end{align*} The next step proves that $P(D)(\chi u)\in H^r(\mathbb{R}^n)$. With that membership in hand, the boundedness of $Q:H^r(\mathbb{R}^n)\to H^{r+m}(\mathbb{R}^n)$ gives $QP(D)(\chi u)\in H^{r+m}(\mathbb{R}^n)$. The smoothing estimate for $R$ with $a=r+m-1$ and $b=r+m$ gives \begin{align*} \|R(\chi u)\|_{H^{r+m}(\mathbb{R}^n)} \leq C_R\|\chi u\|_{H^{r+m-1}(\mathbb{R}^n)}. \end{align*} Since $\chi u=\chi\chi_1u$, the cutoff estimate gives \begin{align*} \|\chi u\|_{H^{r+m-1}(\mathbb{R}^n)} \leq C_\chi\|\chi_1u\|_{H^{r+m-1}(\mathbb{R}^n)}. \end{align*} Therefore, after the next step establishes $P(D)(\chi u)\in H^r(\mathbb{R}^n)$, the identity \begin{align*} \chi u = QP(D)(\chi u) + R(\chi u) \end{align*} holds as an identity of elements of $H^{r+m}(\mathbb{R}^n)$, and the triangle inequality gives \begin{align*} \|\chi u\|_{H^{r+m}(\mathbb{R}^n)} \leq C_2\|P(D)(\chi u)\|_{H^r(\mathbb{R}^n)} + C_3\|\chi_1u\|_{H^{r+m-1}(\mathbb{R}^n)}. \end{align*} [/step]

[step:Expand the commutator and bound its lower-order terms] Because $P(D)$ has constant coefficients, the Leibniz rule gives \begin{align*} P(D)(\chi u)=\chi P(D)u + [P(D),\chi]u \end{align*} in $\mathcal{S}'(\mathbb{R}^n)$, where the commutator $[P(D),\chi]$ is a differential operator of order at most $m-1$ with smooth compactly supported coefficients. More explicitly, there are functions $b_\beta \in C_c^\infty(\mathbb{R}^n)$, indexed by multi-indices $\beta$ with $|\beta|\leq m-1$, such that \begin{align*} [P(D),\chi]u=\sum_{|\beta|\leq m-1} b_\beta D^\beta u. \end{align*} Each coefficient $b_\beta$ is supported in the set where at least one derivative of $\chi$ is nonzero, hence in $\operatorname{supp}\chi$. Since $\chi_1=1$ on a neighbourhood of $\operatorname{supp}\chi$, we have \begin{align*} b_\beta D^\beta u = b_\beta D^\beta(\chi_1u) \end{align*} as distributions. For each $\beta$ with $|\beta|\leq m-1$, differentiation gives a bounded map \begin{align*} D^\beta:H^{r+m-1}(\mathbb{R}^n)\to H^{r+m-1-|\beta|}(\mathbb{R}^n). \end{align*} Since $r+m-1-|\beta|\geq r$, the Sobolev embedding of orders gives \begin{align*} \|D^\beta(\chi_1u)\|_{H^r(\mathbb{R}^n)} \leq C_\beta\|\chi_1u\|_{H^{r+m-1}(\mathbb{R}^n)}. \end{align*} Multiplication by $b_\beta$ is bounded on $H^r(\mathbb{R}^n)$, so summing over the finite set of multi-indices gives \begin{align*} \|[P(D),\chi]u\|_{H^r(\mathbb{R}^n)} \leq C_4\|\chi_1u\|_{H^{r+m-1}(\mathbb{R}^n)}. \end{align*} Also, since $\chi=\chi\chi_1$, the cutoff estimate gives \begin{align*} \|\chi P(D)u\|_{H^r(\mathbb{R}^n)} \leq C_5\|\chi_1P(D)u\|_{H^r(\mathbb{R}^n)}. \end{align*} Combining these two bounds yields \begin{align*} \|P(D)(\chi u)\|_{H^r(\mathbb{R}^n)} \leq C_6\left(\|\chi_1P(D)u\|_{H^r(\mathbb{R}^n)}+\|\chi_1u\|_{H^{r+m-1}(\mathbb{R}^n)}\right). \end{align*} Substituting this into the estimate from the previous step proves \begin{align*} \|\chi u\|_{H^{r+m}(\mathbb{R}^n)} \leq C\left(\|\chi_1P(D)u\|_{H^r(\mathbb{R}^n)}+\|\chi_1u\|_{H^{r+m-1}(\mathbb{R}^n)}\right). \end{align*} This is the asserted a priori estimate. [/step]

[step:Conclude that the a priori estimate already holds for distributions] No separate regularization passage is needed. The operators $Q$, $R$, $P(D)$, and $\psi(D)$ were defined on $\mathcal{S}'(\mathbb{R}^n)$, and all identities used above were identities in $\mathcal{S}'(\mathbb{R}^n)$. The hypotheses $\chi_1P(D)u\in H^r(\mathbb{R}^n)$ and $\chi_1u\in H^{r+m-1}(\mathbb{R}^n)$, together with the cutoff estimate, the commutator expansion, and the Fourier multiplier bounds, imply that both terms \begin{align*} QP(D)(\chi u) \end{align*} and \begin{align*} R(\chi u) \end{align*} belong to $H^{r+m}(\mathbb{R}^n)$. Since the distributional identity \begin{align*} \chi u = QP(D)(\chi u) + R(\chi u) \end{align*} then represents $\chi u$ as a sum of two elements of $H^{r+m}(\mathbb{R}^n)$, it follows that $\chi u\in H^{r+m}(\mathbb{R}^n)$ and the displayed estimate holds for every $u\in\mathcal{S}'(\mathbb{R}^n)$ satisfying the two localized Sobolev hypotheses. [/step]

[step:Bootstrap inward from an outer cutoff to obtain full elliptic regularity] Assume now that $P(D)u=f$ in $\mathcal{S}'(\mathbb{R}^n)$ and $f\in H^s_{\mathrm{loc}}(\mathbb{R}^n)$. Let $K\subset\mathbb{R}^n$ be compact. Choose cutoffs \begin{align*} \chi_0,\chi_1,\dots,\chi_N \in C_c^\infty(\mathbb{R}^n) \end{align*} so that $\chi_0=1$ on a neighbourhood of $K$ and $\chi_{j+1}=1$ on a neighbourhood of $\operatorname{supp}\chi_j$ for every $j\in\{0,\dots,N-1\}$. Since $\chi_Nu$ is a compactly supported distribution, the standard finite-order structure theorem for compactly supported distributions implies that there exists $M>0$ such that \begin{align*} \chi_Nu\in H^{-M}(\mathbb{R}^n). \end{align*} Increase $N$ if necessary so that $N\geq M+s+m$, with the evident interpretation after replacing $M+s+m$ by a larger integer when it is not an integer. We now propagate regularity inward. Define the initial order $t_N=-M$. Suppose that for some $j\in\{0,\dots,N-1\}$ the larger cutoff satisfies \begin{align*} \chi_{j+1}u\in H^{t_{j+1}}(\mathbb{R}^n). \end{align*} Set \begin{align*} r_j=\min\{s,t_{j+1}-m+1\}. \end{align*} Because $f\in H^s_{\mathrm{loc}}(\mathbb{R}^n)$ and $\chi_{j+1}$ is compactly supported, the Sobolev order embedding gives \begin{align*} \chi_{j+1}f\in H^{r_j}(\mathbb{R}^n). \end{align*} Since $P(D)u=f$ in $\mathcal{S}'(\mathbb{R}^n)$, this says \begin{align*} \chi_{j+1}P(D)u\in H^{r_j}(\mathbb{R}^n). \end{align*} Also $r_j+m-1\leq t_{j+1}$, so the induction hypothesis gives \begin{align*} \chi_{j+1}u\in H^{r_j+m-1}(\mathbb{R}^n). \end{align*} Applying the local estimate with the inner cutoff $\chi_j$ and the outer cutoff $\chi_{j+1}$ yields \begin{align*} \chi_ju\in H^{r_j+m}(\mathbb{R}^n). \end{align*} Thus we may define \begin{align*} t_j=r_j+m=\min\{s+m,t_{j+1}+1\}. \end{align*} Starting from $t_N=-M$, this recursion increases the Sobolev order by one at each inward step until it reaches $s+m$. Because $N\geq M+s+m$, after finitely many inward applications we obtain \begin{align*} \chi_0u\in H^{s+m}(\mathbb{R}^n). \end{align*} Since $\chi_0=1$ on a neighbourhood of $K$, this proves $u\in H^{s+m}$ near $K$. The compact set $K$ was arbitrary, hence \begin{align*} u\in H^{s+m}_{\mathrm{loc}}(\mathbb{R}^n). \end{align*} [/step]