Let $n \in \mathbb{N}$ and identify $T^*\mathbb{R}^n$ with $\mathbb{R}^n_x \times \mathbb{R}^n_\xi$. Let $h_0>0$, and let $m_1,m_2:T^*\mathbb{R}^n \to (0,\infty)$ be order functions. Let $r:T^*\mathbb{R}^n \times (0,h_0] \to \mathbb{C}$ and $a:T^*\mathbb{R}^n \times (0,h_0] \to \mathbb{C}$ be functions that are smooth in the variables $(x,\xi)$ for each $h \in (0,h_0]$.

For a positive weight $m:T^*\mathbb{R}^n \to (0,\infty)$, write $b \in S(m)$ to mean that $b:T^*\mathbb{R}^n \times (0,h_0] \to \mathbb{C}$ is smooth in $(x,\xi)$ and, for every pair of multiindices $\alpha,\beta \in \mathbb{N}_0^n$, there is a constant $C_{\alpha,\beta}>0$ such that

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

for all $(x,\xi,h) \in T^*\mathbb{R}^n \times (0,h_0]$.

Assume $r \in h^\infty S(m_1)$, meaning that for every $N \in \mathbb{N}_0$ there exists a function $r_N:T^*\mathbb{R}^n \times (0,h_0] \to \mathbb{C}$ with $r_N \in S(m_1)$ and $r=h^N r_N$. Assume also that $a \in S(m_2)$. Then the pointwise product $ar:T^*\mathbb{R}^n \times (0,h_0] \to \mathbb{C}$ satisfies $ar \in h^\infty S(m_1m_2)$.

Equivalently, for every $N \in \mathbb{N}_0$ there exists a function $c_N:T^*\mathbb{R}^n \times (0,h_0] \to \mathbb{C}$ with $c_N \in S(m_1m_2)$ such that $ar = h^N c_N$.

For a fixed real number $\delta$ satisfying

\begin{align*} 0 \leq \delta < \frac{1}{2}, \end{align*}

write $b \in S_\delta(m)$ to mean that $b:T^*\mathbb{R}^n \times (0,h_0] \to \mathbb{C}$ is smooth in $(x,\xi)$ and, for every pair of multiindices $\alpha,\beta \in \mathbb{N}_0^n$, there is a constant $C_{\alpha,\beta,\delta}>0$ such that

\begin{align*} |\partial_x^\alpha\partial_\xi^\beta b(x,\xi;h)| \leq C_{\alpha,\beta,\delta}h^{-\delta(|\alpha|+|\beta|)}m(x,\xi) \end{align*}

for all $(x,\xi,h) \in T^*\mathbb{R}^n \times (0,h_0]$. If $r \in h^\infty S_\delta(m_1)$ and $a \in S_\delta(m_2)$, then $ar \in h^\infty S_\delta(m_1m_2)$.