Let $K \subset T^*\mathbb{R}^n$ be compact, let $h_0>0$, and let $\chi_1,\dots,\chi_N \in C_c^\infty(T^*\mathbb{R}^n)$ satisfy

\begin{align*} \sum_{j=1}^N \chi_j=1 \end{align*}

on an open neighbourhood $U$ of $K$. For $a\in C_c^\infty(T^*\mathbb{R}^n)$ define the left semiclassical quantization on $\mathcal{S}(\mathbb{R}^n)$ by

\begin{align*} \operatorname{Op}_h(a)v(x)=(2\pi h)^{-n}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi/h}a(x,\xi)v(y)\,dy\,d\xi. \end{align*}

Here $dy$ and $d\xi$ denote Lebesgue measure on the corresponding copies of $\mathbb{R}^n$. For each $j$, put $A_j=\operatorname{Op}_h(\chi_j)$. Let $u_h\in\mathcal{S}'(\mathbb{R}^n)$ be a semiclassically tempered family such that, for one cutoff $\rho\in C_c^\infty(T^*\mathbb{R}^n)$ with $\rho=1$ on a neighbourhood of $K$ and $\operatorname{supp}\rho\subset U$, the remainder

\begin{align*} (I-\operatorname{Op}_h(\rho))u_h \end{align*}

is $O(h^\infty)$ in $H_h^s(\mathbb{R}^n)$ for every $s\in\mathbb{R}$. Then, for every $s\in\mathbb{R}$ and every $M\in\mathbb{N}$, there is a constant $C_{s,M}>0$ such that

\begin{align*} \left\|u_h-\sum_{j=1}^N A_j u_h\right\|_{H_h^s}\le C_{s,M}h^M \end{align*}

for all $0<h\le h_0$. Consequently, for every $s\in\mathbb{R}$ and every $M\in\mathbb{N}$, there are constants $C_s,C_{s,M}>0$ such that

\begin{align*} \|u_h\|_{H_h^s}\le C_s\sum_{j=1}^N\|A_j u_h\|_{H_h^s}+C_{s,M}h^M \end{align*}

for all $0<h\le h_0$.