Let $U \subset \mathbb{R}^n$ be open, let $(m_j)_{j=0}^{\infty}$ be a decreasing sequence of [real numbers](/page/Real%20Numbers) with $m_j \to -\infty$, and for each $j \in \mathbb{N} \cup \{0\}$ let $a_j: U \times \mathbb{R}^n \to \mathbb{C}$ be a complex-valued symbol satisfying $a_j \in S^{m_j}_{1,0}(U \times \mathbb{R}^n)$. Let $a: U \times \mathbb{R}^n \to \mathbb{C}$ be a smooth function, $a \in C^\infty(U \times \mathbb{R}^n;\mathbb{C})$. Suppose that $a$ has the asymptotic symbol expansion $a \sim \sum_{j=0}^{\infty} a_j$, meaning that for every $N \geq 1$, the truncated remainder $a - \sum_{j=0}^{N-1} a_j$ belongs to $S^{m_N}_{1,0}(U \times \mathbb{R}^n)$. For $\xi \in \mathbb{R}^n$, write $\langle \xi \rangle = (1 + |\xi|^2)^{1/2}$. Then for every $N \geq 1$, every compact set $K \subset U$, and every pair of multi-indices $\alpha,\beta \in (\mathbb{N} \cup \{0\})^n$, there exists a constant $C_{K,\alpha,\beta,N} > 0$ such that

\begin{align*} \left|\partial_x^\alpha \partial_\xi^\beta\left(a(x,\xi)-\sum_{j=0}^{N-1}a_j(x,\xi)\right)\right| \leq C_{K,\alpha,\beta,N}\langle \xi\rangle^{m_N-|\beta|} \end{align*}

for all $x \in K$ and all $\xi \in \mathbb{R}^n$.