[proofplan] We prove smoothness locally. On an arbitrary compact set $K \Subset U$, we choose compactly supported cutoffs equal to $1$ near $K$ and apply the local elliptic a priori estimate for $P$. The smoothness of $Pu$ gives finite Sobolev norms of the main term at every order, while the general finite local Sobolev order of distributions controls the lower-order error term. This gives $u \in H^r$ near $K$ for every real $r$, and Sobolev embedding then implies that $u$ is smooth near $K$. Since $K$ was arbitrary, $u$ is smooth on all of $U$. [/proofplan]

[step:Localize near an arbitrary compact subset of $U$] Fix a compact set $K \Subset U$. Choose open sets $V_0,V_1 \subset U$ such that \begin{align*} K \subset V_0 \Subset V_1 \Subset U. \end{align*} Choose cutoff functions \begin{align*} \phi_0: U &\to \mathbb{R} \end{align*} and \begin{align*} \phi_1: U &\to \mathbb{R} \end{align*} with $\phi_0,\phi_1 \in C_c^\infty(U)$, $\phi_0 = 1$ on a neighbourhood of $K$, $\operatorname{supp}\phi_0 \subset V_0$, and $\phi_1 = 1$ on a neighbourhood of $\operatorname{supp}\phi_0$. We view $\phi_0 u$ and $\phi_1 Pu$ as compactly supported distributions on $\mathbb{R}^n$ by extension by zero outside $U$. This is legitimate because $\phi_0$ and $\phi_1$ have compact support in $U$. [/step]

[step:Apply the localized elliptic a priori estimate]Let $s \in \mathbb{R}$ and let $N \in \mathbb{N}$ be a positive integer to be chosen below. For each $t \in \mathbb{R}$, the notation $H^t(\mathbb{R}^n)$ means the $L^2$-[Sobolev space](/page/Sobolev%20Space) of order $t$ on $\mathbb{R}^n$, as declared in the theorem statement. We use the standard localized elliptic a priori estimate for elliptic pseudodifferential operators of order $m$ (citing a result not yet in the wiki: [localized elliptic a priori estimate for pseudodifferential operators](/theorems/7702)). Applied to the cutoffs $\phi_0,\phi_1$ and to the distribution $u \in \mathcal{D}'(U)$, it gives a constant $C = C(P,\phi_0,\phi_1,s,N) > 0$ such that \begin{align*} \|\phi_0 u\|_{H^{s+m}(\mathbb{R}^n)} \leq C\left(\|\phi_1 Pu\|_{H^s(\mathbb{R}^n)} + \|\phi_1 u\|_{H^{-N}(\mathbb{R}^n)}\right). \end{align*} The estimate applies because $P$ is elliptic of order $m$, the cutoffs are compactly supported in $U$, and $\phi_1$ is identically $1$ on a neighbourhood of $\operatorname{supp}\phi_0$.[/step]

[guided]The goal is to turn regularity of $Pu$ into regularity of $u$. The ellipticity of $P$ is exactly the hypothesis that permits this reversal, modulo a lower-order error. We use the following standard local estimate: if $P \in \Psi^m(U)$ is elliptic, if $\phi_0,\phi_1 \in C_c^\infty(U)$, and if $\phi_1 = 1$ near $\operatorname{supp}\phi_0$, then for each $s \in \mathbb{R}$ and each $N \in \mathbb{N}$ there is a constant $C = C(P,\phi_0,\phi_1,s,N) > 0$ such that every $v \in \mathcal{D}'(U)$ satisfies \begin{align*} \|\phi_0 v\|_{H^{s+m}(\mathbb{R}^n)} \leq C\left(\|\phi_1 Pv\|_{H^s(\mathbb{R}^n)} + \|\phi_1 v\|_{H^{-N}(\mathbb{R}^n)}\right). \end{align*} This is the localized elliptic a priori estimate for pseudodifferential operators (citing a result not yet in the wiki: localized elliptic a priori estimate for pseudodifferential operators). We verify the hypotheses before applying it. The operator $P$ is elliptic of order $m$ by assumption. The functions $\phi_0$ and $\phi_1$ lie in $C_c^\infty(U)$ by construction. The support condition $\phi_1 = 1$ near $\operatorname{supp}\phi_0$ also holds by construction. Therefore the estimate applies with $v = u$ and yields \begin{align*} \|\phi_0 u\|_{H^{s+m}(\mathbb{R}^n)} \leq C\left(\|\phi_1 Pu\|_{H^s(\mathbb{R}^n)} + \|\phi_1 u\|_{H^{-N}(\mathbb{R}^n)}\right). \end{align*} The first term measures the known regularity of $Pu$. The second term is the price paid for localization and for the smoothing remainder in the parametrix construction; it only asks for sufficiently negative Sobolev regularity of $u$, which every compactly localized distribution has.[/guided]

[step:Control the term involving $Pu$ by smoothness] By hypothesis, $Pu$ is represented by a function in $C^\infty(U)$. Since $\phi_1 \in C_c^\infty(U)$, the compactly supported function $\phi_1 Pu$ belongs to $C_c^\infty(U)$. Extending by zero to $\mathbb{R}^n$, we obtain \begin{align*} \phi_1 Pu \in C_c^\infty(\mathbb{R}^n). \end{align*} Every compactly supported smooth function on $\mathbb{R}^n$ lies in $H^s(\mathbb{R}^n)$ for every $s \in \mathbb{R}$, so \begin{align*} \|\phi_1 Pu\|_{H^s(\mathbb{R}^n)} < \infty \end{align*} for the chosen $s$. [/step]

[step:Choose a negative Sobolev order that contains the localized distribution] The distribution $\phi_1 u$ has compact support in $\mathbb{R}^n$. By the standard finite Sobolev order property of compactly supported distributions (citing a result not yet in the wiki: compactly supported distributions have finite negative Sobolev order), there exists an integer $N_0 \in \mathbb{N}$ such that \begin{align*} \phi_1 u \in H^{-N_0}(\mathbb{R}^n). \end{align*} Choose $N = N_0$ in the elliptic estimate. Then \begin{align*} \|\phi_1 u\|_{H^{-N}(\mathbb{R}^n)} < \infty. \end{align*} Combining this with the previous step in the elliptic estimate gives \begin{align*} \|\phi_0 u\|_{H^{s+m}(\mathbb{R}^n)} < \infty. \end{align*} Thus $\phi_0 u \in H^{s+m}(\mathbb{R}^n)$. [/step]

[step:Upgrade to every local Sobolev order] Let $r \in \mathbb{R}$ be arbitrary. Set $s = r - m$. The preceding argument applies to this value of $s$ and gives \begin{align*} \phi_0 u \in H^r(\mathbb{R}^n). \end{align*} Since $r$ was arbitrary, \begin{align*} \phi_0 u \in H^r(\mathbb{R}^n) \end{align*} for every $r \in \mathbb{R}$. Because $\phi_0 = 1$ on a neighbourhood of $K$, this means that $u \in H^r$ on a neighbourhood of $K$ for every $r \in \mathbb{R}$. [/step]

[step:Apply Sobolev embedding to obtain smoothness near $K$]Let $\alpha = (\alpha_1,\dots,\alpha_n) \in \mathbb{N}_0^n$ be an arbitrary multi-index, and let $|\alpha| = \alpha_1+\cdots+\alpha_n$. Choose $r_\alpha \in \mathbb{R}$ such that \begin{align*} r_\alpha > |\alpha| + \frac{n}{2}. \end{align*} From the previous step, $\phi_0 u \in H^{r_\alpha}(\mathbb{R}^n)$. By the [Sobolev embedding theorem](/theorems/903) on $\mathbb{R}^n$ (citing a result not yet in the wiki: Sobolev embedding theorem), $H^{r_\alpha}(\mathbb{R}^n)$ embeds continuously into $C^{|\alpha|}(\mathbb{R}^n)$. Therefore $\phi_0 u$ is represented by a $C^{|\alpha|}$ function on $\mathbb{R}^n$. Since $\phi_0 = 1$ on a neighbourhood of $K$, the distribution $u$ is represented by a $C^{|\alpha|}$ function on a neighbourhood of $K$. The multi-index $\alpha$ was arbitrary, so $u$ is represented by a $C^\infty$ function on a neighbourhood of $K$.[/step]

[guided]We now convert Sobolev regularity of all orders into classical smoothness. Fix a multi-index $\alpha = (\alpha_1,\dots,\alpha_n) \in \mathbb{N}_0^n$, and define its order by $|\alpha| = \alpha_1+\cdots+\alpha_n$. To prove smoothness, it is enough to prove that derivatives of every order exist continuously near $K$. Choose a real number $r_\alpha$ satisfying \begin{align*} r_\alpha > |\alpha| + \frac{n}{2}. \end{align*} The previous step gives $\phi_0 u \in H^{r_\alpha}(\mathbb{R}^n)$. We apply the Sobolev embedding theorem on $\mathbb{R}^n$ (citing a result not yet in the wiki: Sobolev embedding theorem). Its hypothesis is precisely that the Sobolev exponent exceeds the desired number of continuous derivatives by $n/2$; here this is the inequality $r_\alpha > |\alpha| + n/2$. Hence \begin{align*} H^{r_\alpha}(\mathbb{R}^n) \hookrightarrow C^{|\alpha|}(\mathbb{R}^n). \end{align*} It follows that $\phi_0 u$ is represented by a $C^{|\alpha|}$ function on $\mathbb{R}^n$. Why does this say something about $u$ rather than only about $\phi_0 u$? The cutoff was chosen so that $\phi_0 = 1$ on a neighbourhood of $K$. On that neighbourhood, multiplication by $\phi_0$ does not change the distribution $u$. Therefore $u$ itself is represented by a $C^{|\alpha|}$ function near $K$. Since $\alpha$ was arbitrary, this holds for every derivative order, and $u$ is smooth near $K$.[/guided]

[step:Conclude smoothness on all of $U$] We have shown that for every compact set $K \Subset U$, the distribution $u$ is represented by a smooth function on a neighbourhood of $K$. For each point $x \in U$, choose $K = \overline{B}(x,\rho)$ with $\rho > 0$ small enough that $K \Subset U$. The preceding conclusion gives a neighbourhood of $x$ on which $u$ is smooth. Hence $u \in C^\infty(U)$ as a distribution represented locally by smooth functions. This proves the theorem. [/step]