denote its semiclassical quantization for every $r\in\mathbb{R}$, with the same quantization convention used in the semiclassical symbolic calculus. Let
belong to $C_c^\infty(T^*\mathbb{R}^n)$, and assume $\chi_0\prec\chi_1$, meaning that $\chi_1=1$ on an open neighbourhood of $\operatorname{supp}\chi_0$. Suppose that $p$ is uniformly elliptic on an open neighbourhood of $\operatorname{supp}\chi_1$: there are an open set $U\subset T^*\mathbb{R}^n$ with $\operatorname{supp}\chi_1\subset U$ and constants $c>0$, $h_1\in(0,h_0]$ such that
for every $r\in\mathbb{R}$. Then for every $s\in\mathbb{R}$ and every $M\in\mathbb{N}$ there exist constants $C_{s,M}>0$ and $N_{s,M}\in\mathbb{N}$ such that, for every $0<h\leq h_1$ and every $u_h\in H_h^{-N_{s,M}}(\mathbb{R}^n)$ for which $P_hu_h$ is defined as a distribution and $A_1P_hu_h\in H_h^s(\mathbb{R}^n)$, one has