[proofplan] The proof is a localization argument. Each summand is first interpreted as a Euclidean classical pseudodifferential operator transported through a chart and cut off near the diagonal. Local finiteness makes the kernel sum meaningful, while the assumed properness of the union of supports gives proper support. The overlap assertions follow from the scalar coordinate-change formula for pseudodifferential symbols: agreement of principal terms lowers the order by one, and agreement of the complete transformed expansion leaves only a smoothing remainder. Finally, Borel summation for classical symbols realizes any compatible sequence of homogeneous pieces by genuine local symbols. [/proofplan]

[step:Build the localized kernels in coordinates] Fix $i \in I$. Define the coordinate pullback map $\kappa_i^*: C^\infty(V_i) \to C^\infty(U_i)$ by $\kappa_i^* f=f\circ\kappa_i$, and define the coordinate pushforward map $(\kappa_i)_*: C_c^\infty(U_i) \to C_c^\infty(V_i)$ by $(\kappa_i)_*u=u\circ\kappa_i^{-1}$. With the scalar-function convention fixed in the statement, $\operatorname{Op}_{\rho_i}(a_i)$ is the operator on $C_c^\infty(U_i)$ obtained by pushing a [test function](/page/Test%20Function) forward to $V_i$, applying the Euclidean scalar quantization with [Lebesgue measure](/page/Lebesgue%20Measure) $d\mathcal{L}^n$, multiplying the Euclidean kernel by $\rho_i\circ(\kappa_i^{-1}\times\kappa_i^{-1})$, and pulling the result back to $U_i$. Equivalently, for $u \in C_c^\infty(U_i)$ and $p \in U_i$, one first sets $\tilde u := (\kappa_i)_*u \in C_c^\infty(V_i)$, applies the Euclidean oscillatory operator to $\tilde u$, and then evaluates the resulting function at $\kappa_i(p)$. Its Euclidean kernel, before transport back to $U_i \times U_i$, is given near the diagonal by the oscillatory formula \begin{align*} K_{a_i,\rho_i}^{\mathrm{euc}}(x,y)=(2\pi)^{-n}\int_{\mathbb{R}^n} e^{i(x-y)\cdot \xi}\rho_i(\kappa_i^{-1}(x),\kappa_i^{-1}(y))a_i(x,\xi)\,d\mathcal{L}^n(\xi). \end{align*} Here $x,y \in V_i$, and the integral is understood as an oscillatory integral. The [open set](/page/Open%20Set) $O_i$ is used at this point: whenever $p \in \operatorname{supp}\psi_i$ and $q \in \operatorname{supp}\chi_i$ are sufficiently close to the diagonal, $(p,q) \in O_i$, hence $\rho_i(p,q)=1$ and the singular near-diagonal Euclidean pseudodifferential kernel is unchanged by the diagonal cutoff. Let $K_i \in \mathcal{D}'(U_i \times U_i)$ denote the pullback of this Euclidean kernel by $\kappa_i \times \kappa_i$. The kernel of the localized summand \begin{align*} A_i := \psi_i\operatorname{Op}_{\rho_i}(a_i)\chi_i \end{align*} is therefore the distribution \begin{align*} L_i(p,q)=\psi_i(p)K_i(p,q)\chi_i(q) \end{align*} on $U_i \times U_i$, extended by zero to $M \times M$. By hypothesis, $\operatorname{supp}L_i$ lies in the coordinate neighbourhood where the oscillatory formula is valid. [/step]

[step:Sum the locally finite kernels and prove proper support]Let $\mathcal{D}'(M \times M)$ denote the space of distributions on $M \times M$, that is, the continuous linear functionals on $C_c^\infty(M \times M)$. Define $K_A \in \mathcal{D}'(M \times M)$ by \begin{align*} K_A := \sum_{i \in I} L_i. \end{align*} This sum is well-defined as a distribution because the atlas is locally finite and because each $L_i$ is supported in $U_i \times U_i$ after multiplication by compactly supported cutoffs. To see this explicitly, let $\Phi \in C_c^\infty(M \times M)$ be a test function. Since $\operatorname{supp}\Phi$ is compact and the family $\{U_i\}_{i \in I}$ is locally finite, only finitely many sets $U_i \times U_i$ meet $\operatorname{supp}\Phi$. Hence only finitely many pairings $L_i(\Phi)$ are nonzero, and the formula \begin{align*} K_A(\Phi):=\sum_{i \in I}L_i(\Phi) \end{align*} defines a continuous linear functional on $C_c^\infty(M \times M)$. Let \begin{align*} S:=\bigcup_{i \in I}\operatorname{supp}L_i \subset M \times M. \end{align*} Let $\pi_1: M \times M \to M$ and $\pi_2: M \times M \to M$ denote the first and second coordinate projections, so $\pi_1(p,q)=p$ and $\pi_2(p,q)=q$. By the support hypothesis in the statement, both restricted projections \begin{align*} \pi_1|_S:S\to M \end{align*} and \begin{align*} \pi_2|_S:S\to M \end{align*} are proper. Since $\operatorname{supp}K_A \subset S$, the restrictions of $\pi_1$ and $\pi_2$ to $\operatorname{supp}K_A$ are also proper. Thus the operator with Schwartz kernel $K_A$ is properly supported.[/step]

[guided]We need two separate facts: first, that the infinite sum of kernels is actually meaningful, and second, that the resulting operator is properly supported. Start with the distributional sum. Let $\Phi \in C_c^\infty(M \times M)$ be a test function. The compact set $\operatorname{supp}\Phi$ can meet only finitely many products $U_i \times U_i$, because the atlas $\{U_i\}_{i \in I}$ is locally finite. Since $L_i$ is supported in $U_i \times U_i$, all but finitely many numbers $L_i(\Phi)$ vanish. Therefore \begin{align*} K_A(\Phi):=\sum_{i \in I}L_i(\Phi) \end{align*} is a finite sum for each test function $\Phi$. This proves that $K_A$ is a well-defined distribution on $M \times M$. Now define the set containing all possible kernel support: \begin{align*} S:=\bigcup_{i \in I}\operatorname{supp}L_i. \end{align*} Let $\pi_1: M \times M \to M$ and $\pi_2: M \times M \to M$ be the first and second coordinate projections, given by $\pi_1(p,q)=p$ and $\pi_2(p,q)=q$. The theorem assumes that the union of the local kernel supports has proper projections to $M$. This means that, for every compact set $C \subset M$, both inverse images \begin{align*} (\pi_1|_S)^{-1}(C) \quad \text{and} \quad (\pi_2|_S)^{-1}(C) \end{align*} are compact subsets of $S$. Since the support of the summed kernel satisfies $\operatorname{supp}K_A \subset S$, the projections restricted to $\operatorname{supp}K_A$ are proper as well. This is exactly the proper support condition for the operator defined by $K_A$: compact sets in the input or output variable interact with only compact portions of the kernel.[/guided]

[step:Identify the summed operator as locally classical of order $m$] Let $W \subset M$ be a coordinate neighbourhood whose closure meets only finitely many $U_i$. Such $W$ exists locally by local finiteness. In the chart on $W$, the restriction of $A$ is a finite sum of transported Euclidean operators \begin{align*} \psi_i\operatorname{Op}_{\rho_i}(a_i)\chi_i. \end{align*} Each transition map between the chart on $W$ and a chart $(U_i,\kappa_i)$ is a smooth diffeomorphism between open subsets of $\mathbb{R}^n$. The transported kernel is an oscillatory kernel with smooth amplitude compactly supported in the two spatial variables, because $\psi_i$ and $\chi_i$ are compactly supported, and classical of order $m$ in the covariable, because $a_i \in S_{\mathrm{cl}}^m(V_i \times \mathbb{R}^n)$. The standard scalar coordinate-change theorem for Euclidean pseudodifferential symbols applies to this transported amplitude: its hypotheses are the smooth diffeomorphic coordinate change, the scalar convention using Lebesgue measure in both coordinate systems, compact spatial support, and the classical symbol estimates in $\xi$. In the scalar convention, the Jacobian factor introduced by changing the integration variable in the base variable is part of the transformed amplitude; the amplitude-to-symbol expansion then produces a Euclidean classical symbol whose leading homogeneous term is obtained by the cotangent transformation, while all determinant and derivative contributions enter only through the prescribed lower-order terms of the scalar coordinate-change formula. Multiplication by the smooth compactly supported functions $\psi_i$ and $\chi_i$ preserves the Euclidean classical pseudodifferential order. Because $O_i$ contains the diagonal over $\operatorname{supp}\psi_i\cap\operatorname{supp}\chi_i$ and $\rho_i=1$ on $O_i$, the singular kernel on that near-diagonal region is exactly the original Euclidean pseudodifferential kernel. On the complement of $O_i$ inside the spatial support of $\psi_i(p)\chi_i(q)$, the points $p$ and $q$ are separated from the diagonal; in coordinates this means $x-y$ is bounded away from $0$ on that compact spatial support. Therefore the phase $(x-y)\cdot\xi$ has no stationary point in $\xi$. Repeated [integration by parts](/theorems/210) with the operator determined by $\nabla_\xi((x-y)\cdot\xi)=x-y$ gives a smooth kernel. Hence each summand is a classical pseudodifferential operator of order $m$ in local coordinates. A finite sum of Euclidean classical pseudodifferential operators of order $m$ is again a Euclidean classical pseudodifferential operator of order $m$. Therefore $A$ is locally a classical pseudodifferential operator of order $m$ and, by the previous step, properly supported. Thus \begin{align*} A \in \Psi_{\mathrm{cl}}^m(M). \end{align*} [/step]

[step:Compare principal symbols across coordinate overlaps]Let $i,j \in I$ with $U_i \cap U_j \neq \varnothing$. Write \begin{align*} F_{ij}: \kappa_i(U_i \cap U_j)\to \kappa_j(U_i \cap U_j), \qquad F_{ij}=\kappa_j\circ\kappa_i^{-1}. \end{align*} For the scalar-function convention fixed in the statement, the coordinate invariance theorem for pseudodifferential symbols applies to the transition $F_{ij}$ because $F_{ij}$ is a smooth diffeomorphism between open subsets of $\mathbb{R}^n$, the transported amplitudes have compact spatial support after multiplication by $\psi_i$ and $\chi_i$, the quantization convention uses Lebesgue measure rather than half-densities, and the diagonal cutoffs equal $1$ on the relevant near-diagonal region. More explicitly, transporting a scalar kernel from the $j$-chart to the $i$-chart introduces the Jacobian determinant from the coordinate change in the base integration variable; the scalar coordinate-change symbol formula incorporates that determinant into the transformed amplitude, and the amplitude-to-symbol expansion places its effect in the full lower-order expansion. Its leading homogeneous term is therefore \begin{align*} (x,\xi)\mapsto a_{j,m}\bigl(F_{ij}(x),(JF_{ij,x}^{-1})^\top \xi\bigr). \end{align*} The cotangent transition law in the statement therefore says that the local degree $m$ terms describe one globally compatible principal symbol, so the order $m$ comparison is independent of which overlap chart is used. Let $a_i$ and $b_i$ be the two local symbol families in the statement, and write $A$ and $B$ for the corresponding patched operators. Since $a_{i,m}=b_{i,m}$ for every $i\in I$, the actual $i$th localized difference has vanishing homogeneous component of degree $m$ in any chart, with the preceding coordinate-change formula ensuring the same conclusion after transport across overlaps. Hence $a_i-b_i\in S_{\mathrm{cl}}^{m-1}(V_i\times\mathbb{R}^n)$. Therefore the $i$th localized difference \begin{align*} \psi_i\operatorname{Op}_{\rho_i}(a_i-b_i)\chi_i \end{align*} is a classical pseudodifferential operator of order at most $m-1$ by the same local-coordinate argument used above. In any coordinate neighbourhood meeting only finitely many $U_i$, the full local representative of $A-B$ is the finite sum of these order $m-1$ localized differences. Thus its principal order $m$ contribution is zero after the actual cutoff-weighted summation, not merely on each overlap before summing. The cutoff pieces supported away from the diagonal have smooth kernels, hence are smoothing and in particular of order $m-1$. Since the localization is locally finite and properly supported, the global difference lies in \begin{align*} \Psi_{\mathrm{cl}}^{m-1}(M). \end{align*}[/step]

[guided]We compare the two patched operators after the localization has been applied, because the global principal symbol of a patched operator is obtained from the cutoff-weighted local summands. Let $A$ be built from $a_i$ and let $B$ be built from $b_i$, using the same atlas, cutoffs, diagonal cutoffs, open sets, and scalar quantization convention. First we verify that the principal-symbol comparison is independent of the overlap chart. On an overlap $U_i\cap U_j$, the transition map \begin{align*} F_{ij}:\kappa_i(U_i\cap U_j)\to\kappa_j(U_i\cap U_j) \end{align*} is a smooth diffeomorphism. The transported localized amplitudes have compact spatial support because the cutoffs $\psi_i$ and $\chi_i$ are compactly supported. The diagonal cutoffs equal $1$ on the near-diagonal region where the singular part of the kernel lies, and the off-diagonal part is smooth by nonstationary phase. Thus the scalar coordinate-change theorem applies. Under the scalar Lebesgue convention, the coordinate change contributes a Jacobian determinant to the transformed amplitude, and the amplitude-to-symbol expansion puts that determinant and its derivatives into the complete lower-order symbol formula. The leading homogeneous term is \begin{align*} (x,\xi)\mapsto a_{j,m}\bigl(F_{ij}(x),(JF_{ij,x}^{-1})^\top\xi\bigr). \end{align*} The cotangent transition law in the statement therefore says that the degree $m$ pieces define one compatible principal symbol. For each $i\in I$, the statement assumes \begin{align*} a_{i,m}=b_{i,m}. \end{align*} This equality says that the order $m$ homogeneous part of the actual $i$th localized difference is zero. Hence \begin{align*} a_i-b_i\in S_{\mathrm{cl}}^{m-1}(V_i\times\mathbb{R}^n). \end{align*} Applying the local construction to the symbol $a_i-b_i$, the operator \begin{align*} \psi_i\operatorname{Op}_{\rho_i}(a_i-b_i)\chi_i \end{align*} is a classical pseudodifferential operator of order at most $m-1$. The hypotheses needed here are the same as before: the transition maps are smooth diffeomorphisms, the cutoffs are smooth and compactly supported, and $O_i$ contains the relevant diagonal region on which $\rho_i=1$, so the singular pseudodifferential kernel is preserved while the off-diagonal cutoff contribution is smooth. Now take any coordinate neighbourhood whose closure meets only finitely many $U_i$. On that neighbourhood, $A-B$ is a finite sum of the localized differences above. A finite sum of classical pseudodifferential operators of order at most $m-1$ is again of order at most $m-1$. This proves cancellation of the order $m$ contribution for the actual patched operator, including the cutoff weights. Proper support follows from the already established properness of the kernel-support union, so \begin{align*} A-B\in\Psi_{\mathrm{cl}}^{m-1}(M). \end{align*}[/guided]

[step:Use complete compatibility to reduce each localized difference to smoothing order] Assume now that two local classical symbol families $a_i,\tilde a_i \in S_{\mathrm{cl}}^m(V_i \times \mathbb{R}^n)$ realize the same complete compatible classical asymptotic symbol. For each fixed $i \in I$, this means that both $a_i$ and $\tilde a_i$ have the same homogeneous terms $a_{i,m-k}$ in their classical expansions for every $k \in \mathbb{N}_0$. Hence their difference has zero homogeneous component of every degree $m-k$, so \begin{align*} a_i-\tilde a_i \in S^{-\infty}(V_i \times \mathbb{R}^n):=\bigcap_{N \in \mathbb{R}}S^N(V_i \times \mathbb{R}^n). \end{align*} By the smoothing criterion for $S^{-\infty}$ symbols, the Euclidean operator with symbol $a_i-\tilde a_i$ has a smooth Schwartz kernel. Multiplication by $\psi_i$ and $\chi_i$, multiplication by the smooth diagonal cutoff $\rho_i$, extension by zero, and transport through the smooth chart $\kappa_i$ preserve smoothness of the kernel. Therefore each localized summand difference \begin{align*} \psi_i\operatorname{Op}_{\rho_i}(a_i-\tilde a_i)\chi_i \end{align*} is smoothing. Since the construction is locally finite and the same proper support hypothesis controls the union of supports, the global difference of the two patched operators is a properly supported smoothing operator on $M$. [/step]

[step:Realize compatible homogeneous data by Borel summation] Let $\{a_{i,m-k}\}_{i \in I,k\in\mathbb{N}_0}$ be a complete compatible classical asymptotic symbol in the sense of the statement. For each fixed $i \in I$, the functions $a_{i,m-k}$ satisfy the classical homogeneous symbol estimates on compact subsets of $V_i$ for $|\xi|\geq1$. Choose a smooth cutoff $\theta:\mathbb{R}^n\to[0,1]$ with $\theta(\xi)=0$ for $|\xi|\leq1$ and $\theta(\xi)=1$ for $|\xi|\geq2$. The products $\theta(\xi)a_{i,m-k}(x,\xi)$ extend the homogeneous pieces across the region near $\xi=0$ as smooth symbols of order $m-k$ on $V_i\times\mathbb{R}^n$. Indeed, for $|\xi|\geq2$ the required bounds are exactly the stated estimates. On the annulus $1<|\xi|<2$, Leibniz's rule gives \begin{align*} \partial_x^\alpha\partial_\xi^\beta(\theta a_{i,m-k})(x,\xi)=\sum_{\gamma\leq\beta}{\beta\choose\gamma}(\partial_\xi^\gamma\theta)(\xi)(\partial_x^\alpha\partial_\xi^{\beta-\gamma}a_{i,m-k})(x,\xi). \end{align*} For each $\gamma\leq\beta$, the derivative $\partial_\xi^\gamma\theta$ is bounded on $\mathbb{R}^n$, and $|\xi|$ remains in the compact annulus $1<|\xi|<2$. The stated estimates for $a_{i,m-k}$ therefore bound each summand by a constant depending only on $i$, $K$, $k$, $\alpha$, $\beta$, $\gamma$, and $\theta$, multiplied by $(1+|\xi|)^{m-k-|\beta|}$ after increasing the constant on the annulus. Summing over the finitely many $γ\leq\beta$ gives the locally uniform symbol bound required for Borel summation on every compact $K\subset V_i$. By Borel summation for classical symbols, there exists a symbol \begin{align*} a_i \in S_{\mathrm{cl}}^m(V_i \times \mathbb{R}^n) \end{align*} such that \begin{align*} a_i \sim \sum_{k=0}^{\infty}a_{i,m-k} \end{align*} as a classical asymptotic expansion. Using these symbols in the fixed localization data gives a patched operator \begin{align*} A=\sum_{i\in I}\psi_i\operatorname{Op}_{\rho_i}(a_i)\chi_i. \end{align*} The first three steps show that $A \in \Psi_{\mathrm{cl}}^m(M)$ and is properly supported. If $\tilde a_i \in S_{\mathrm{cl}}^m(V_i \times \mathbb{R}^n)$ is another family with the same complete compatible asymptotic expansion, then \begin{align*} a_i-\tilde a_i \in S^{-\infty}(V_i \times \mathbb{R}^n) \end{align*} for every $i \in I$. The preceding step therefore implies that the two patched operators differ by a smoothing operator. Hence the complete compatible classical asymptotic symbol is realized and determines the patched operator uniquely modulo smoothing operators. [/step]

[step:Conclude the construction and uniqueness assertions] Combining the local kernel construction, the distributional summation argument, and properness of the support projections proves that \begin{align*} A=\sum_{i \in I}\psi_i\operatorname{Op}_{\rho_i}(a_i)\chi_i \end{align*} is a properly supported classical pseudodifferential operator of order $m$ on $M$. The principal-symbol comparison shows that compatible principal homogeneous parts determine the patched operator modulo $\Psi_{\mathrm{cl}}^{m-1}(M)$. The complete-symbol comparison shows that compatible full classical asymptotic data determine it modulo smoothing operators, and Borel summation realizes every such compatible family. This proves all assertions of the theorem. [/step]