[proofplan] The proof is the standard parametrix argument. Ellipticity of $A$ gives a pseudodifferential operator $B$ of order $-m$ such that $BA$ differs from the identity by a smoothing operator. Applying this identity to $u$ decomposes $u$ as the sum of a Sobolev-regular term $B(Au)$ and a smooth term. The Sobolev mapping theorem for pseudodifferential operators handles the first term, while the smoothing property handles the second. [/proofplan] [step:Choose a parametrix whose error is smoothing] By the parametrix theorem for elliptic classical pseudodifferential operators on closed manifolds, applied to the [elliptic operator](/page/Elliptic%20Operator) $A \in \Psi^m_{\mathrm{cl}}(M)$, there exist operators \begin{align*} B \in \Psi^{-m}_{\mathrm{cl}}(M) \end{align*} and \begin{align*} R \in \Psi^{-\infty}(M) \end{align*} such that, as operators on scalar distributions, \begin{align*} BA = I_{\mathcal{D}'(M)} - R. \end{align*} Here $I_{\mathcal{D}'(M)}: \mathcal{D}'(M) \to \mathcal{D}'(M)$ denotes the identity map, and $R: \mathcal{D}'(M) \to C^\infty(M)$ is smoothing. This invocation uses the defining hypothesis that $A$ is elliptic and classical on the closed manifold $M$. [guided] The purpose of ellipticity is precisely to allow inversion modulo smoothing errors. Since $A \in \Psi^m_{\mathrm{cl}}(M)$ is elliptic and $M$ is closed, the parametrix theorem for elliptic classical pseudodifferential operators on closed manifolds gives an operator \begin{align*} B \in \Psi^{-m}_{\mathrm{cl}}(M) \end{align*} and a smoothing remainder \begin{align*} R \in \Psi^{-\infty}(M) \end{align*} such that \begin{align*} BA = I_{\mathcal{D}'(M)} - R \end{align*} on $\mathcal{D}'(M)$. The sign convention is immaterial, but fixing it now prevents ambiguity later: this identity will become $u = B(Au) + Ru$ after applying it to $u$. The operator $I_{\mathcal{D}'(M)}: \mathcal{D}'(M) \to \mathcal{D}'(M)$ is the identity map, and the notation $R \in \Psi^{-\infty}(M)$ means that $R$ is smoothing, hence maps every distribution on $M$ to a smooth scalar function on $M$. [/guided] [/step] [step:Rewrite $u$ using the parametrix identity] Apply the operator identity $BA = I_{\mathcal{D}'(M)} - R$ to the given distribution $u \in \mathcal{D}'(M)$. Since $Au$ is defined as a scalar distribution and satisfies $Au \in H^s(M)$ by hypothesis, we obtain \begin{align*} B(Au) = u - Ru. \end{align*} Rearranging in the [vector space](/page/Vector%20Space) $\mathcal{D}'(M)$ gives \begin{align*} u = B(Au) + Ru. \end{align*} [/step] [step:Place $B(Au)$ in $H^{s+m}(M)$] The Sobolev mapping theorem for pseudodifferential operators on closed manifolds states that every operator $P \in \Psi^r_{\mathrm{cl}}(M)$ extends continuously as a map \begin{align*} P: H^t(M) \to H^{t-r}(M) \end{align*} for all $t,r \in \mathbb{R}$. Applying this with $P = B$, $r = -m$, and $t = s$, we get a continuous map \begin{align*} B: H^s(M) \to H^{s+m}(M). \end{align*} Because $Au \in H^s(M)$, it follows that \begin{align*} B(Au) \in H^{s+m}(M). \end{align*} [/step] [step:Place the smoothing remainder in every Sobolev space] Since $R \in \Psi^{-\infty}(M)$, the smoothing-operator theorem gives \begin{align*} R: \mathcal{D}'(M) \to C^\infty(M). \end{align*} Thus $Ru \in C^\infty(M)$. On a closed smooth manifold, every smooth scalar function belongs to every [Sobolev space](/page/Sobolev%20Space) in the standard Sobolev scale, so in particular \begin{align*} Ru \in H^{s+m}(M). \end{align*} [/step] [step:Conclude by the linearity of the Sobolev space] The space $H^{s+m}(M)$ is a vector space. From the previous two steps, \begin{align*} B(Au) \in H^{s+m}(M) \end{align*} and \begin{align*} Ru \in H^{s+m}(M). \end{align*} Using the decomposition \begin{align*} u = B(Au) + Ru, \end{align*} we conclude that \begin{align*} u \in H^{s+m}(M). \end{align*} This proves the asserted elliptic regularity statement. [/step]