Let $n \in \mathbb{N}$, and identify $T^*\mathbb{R}^n$ with $\mathbb{R}^n_x \times \mathbb{R}^n_\xi$. Let $m_1,m_2: T^*\mathbb{R}^n \to (0,\infty)$ be order functions. Thus, for each $j \in \{1,2\}$, there exist constants $C_j > 0$ and $N_j \in \mathbb{N}$ such that, for all $\rho,\rho' \in T^*\mathbb{R}^n$,

\begin{align*} m_j(\rho) \leq C_j \langle \rho-\rho' \rangle^{N_j} m_j(\rho'), \end{align*}

where $\langle z\rangle := (1+|z|^2)^{1/2}$.

For an order function $m: T^*\mathbb{R}^n \to (0,\infty)$, define $S(m)$ to be the class of all functions $a \in C^\infty(T^*\mathbb{R}^n;\mathbb{C})$ such that, for every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$, there exists a constant $C_{\alpha,\beta} > 0$ satisfying

\begin{align*} |\partial_x^\alpha \partial_\xi^\beta a(x,\xi)| \leq C_{\alpha,\beta} m(x,\xi) \end{align*}

for all $(x,\xi) \in T^*\mathbb{R}^n$.

For each $q \in \mathbb{R}$, define $m_q: T^*\mathbb{R}^n \to (0,\infty)$ by $m_q(x,\xi) := \langle \xi\rangle^q$, where $\langle \xi\rangle := (1+|\xi|^2)^{1/2}$. The classical symbol class is $S^q(T^*\mathbb{R}^n) := S(m_q)$.

Then:

1. If $a \in S(m_1)$, then $\partial_x^\alpha \partial_\xi^\beta a \in S(m_1)$ for every $\alpha,\beta \in \mathbb{N}_0^n$. 2. If $a \in S(m_1)$ and $b \in S(m_2)$, then $m_1m_2$ is an order function and $ab \in S(m_1m_2)$. 3. If $r,s \in \mathbb{R}$, $a \in S^r(T^*\mathbb{R}^n)$, and $b \in S^s(T^*\mathbb{R}^n)$, then $ab \in S^{r+s}(T^*\mathbb{R}^n)$.