Let $M$ be a smooth $n$-manifold, let $m \in \mathbb{R}$, and fix the scalar-function convention in which every local quantization is computed in coordinates using [Lebesgue measure](/page/Lebesgue%20Measure). Let $\{(U_i,\kappa_i)\}_{i \in I}$ be a locally finite smooth atlas, where each coordinate map $\kappa_i: U_i \to V_i \subset \mathbb{R}^n$ is a diffeomorphism onto an [open set](/page/Open%20Set).

For each $i \in I$, let $\chi_i,\psi_i \in C_c^\infty(U_i)$ and let $\rho_i \in C_c^\infty(U_i \times U_i)$. Assume that there is an open set $O_i \subset U_i \times U_i$ containing the diagonal over $\operatorname{supp}\psi_i \cap \operatorname{supp}\chi_i$ such that $\rho_i = 1$ on $O_i$. For $a_i \in S_{\mathrm{cl}}^m(V_i \times \mathbb{R}^n)$, define $\operatorname{Op}_{\rho_i}(a_i)$ on $C_c^\infty(U_i)$ by transporting to $V_i$, applying the scalar Euclidean oscillatory integral with Lebesgue measure $d\mathcal{L}^n$, multiplying the Euclidean kernel by $\rho_i \circ (\kappa_i^{-1} \times \kappa_i^{-1})$, and transporting back to $U_i$. Assume that the Schwartz kernel of each localized operator $\psi_i \operatorname{Op}_{\rho_i}(a_i)\chi_i$ is supported in the coordinate neighbourhood on which the corresponding oscillatory formula is valid, and that the union of these kernel supports has proper projections to $M$.

For each $i \in I$, let $a_i \in S_{\mathrm{cl}}^m(V_i \times \mathbb{R}^n)$ be a classical Euclidean symbol. Then the locally finite sum

\begin{align*} A := \sum_{i \in I} \psi_i \operatorname{Op}_{\rho_i}(a_i)\chi_i \end{align*}

defines a properly supported classical pseudodifferential operator of order $m$ on scalar functions on $M$.

For $i,j \in I$ with $U_i \cap U_j \neq \varnothing$, define the coordinate transition map

\begin{align*} F_{ij}: \kappa_i(U_i \cap U_j) \to \kappa_j(U_i \cap U_j) \end{align*}

by $F_{ij} := \kappa_j \circ \kappa_i^{-1}$, and let $JF_{ij,x}$ denote the Jacobian matrix of $F_{ij}$ at $x$. Suppose $a_i,b_i \in S_{\mathrm{cl}}^m(V_i \times \mathbb{R}^n)$ are two local symbol families used with the same atlas, cutoffs, diagonal cutoffs, open sets, and scalar quantization convention. If their principal homogeneous components satisfy

\begin{align*} a_{i,m}=b_{i,m} \end{align*}

for every $i \in I$, and the common components satisfy the cotangent transition law

\begin{align*} a_{i,m}(x,\xi)=a_{j,m}(F_{ij}(x),(JF_{ij,x}^{-1})^\top \xi) \end{align*}

for every $x \in \kappa_i(U_i \cap U_j)$ and every $\xi \in \mathbb{R}^n \setminus \{0\}$, then the two resulting patched operators are equal modulo $\Psi_{\mathrm{cl}}^{m-1}(M)$.

A complete compatible classical asymptotic symbol consists of functions $a_{i,m-k}$, indexed by $i \in I$ and $k \in \mathbb{N}_0$, defined for $|\xi| \geq 1$, smooth on $V_i \times (\mathbb{R}^n \setminus \{0\})$, homogeneous of degree $m-k$ in $\xi$ for $|\xi| \geq 1$, and satisfying the following estimates: for every compact set $K \subset V_i$ and every pair of multiindices $\alpha,\beta$, there exists a constant $C_{i,K,k,\alpha,\beta}>0$ such that

\begin{align*} |\partial_x^\alpha \partial_\xi^\beta a_{i,m-k}(x,\xi)| \leq C_{i,K,k,\alpha,\beta}(1+|\xi|)^{m-k-|\beta|} \end{align*}

for all $x \in K$ and all $|\xi| \geq 1$. Compatibility means that, on every overlap $U_i \cap U_j$, the complete expansion obtained from

\begin{align*} \sum_{k \geq 0}a_{j,m-k} \end{align*}

by the scalar coordinate-change symbol formula agrees term by term with

\begin{align*} \sum_{k \geq 0}a_{i,m-k}, \end{align*}

including all lower-order correction terms produced by that formula.

If two local classical symbol families realize the same complete compatible classical asymptotic symbol, then the corresponding patched operators are equal modulo smoothing operators. Conversely, every complete compatible classical asymptotic symbol is realized by local symbols $a_i \in S_{\mathrm{cl}}^m(V_i \times \mathbb{R}^n)$ by Borel summation after inserting a fixed smooth cutoff in the region $|\xi|<1$, and, for the fixed atlas, cutoffs, diagonal cutoffs, open sets, and scalar quantization convention, the resulting patched operator is unique modulo smoothing operators.