Let $n \in \mathbb{N}$ and let $h_0 > 0$. For each $h \in (0,h_0]$, let $a_h: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ be the smooth map $(x,\xi) \mapsto a(x,\xi;h)$, and suppose that for every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$ the uniform seminorm
is finite. For each $h \in (0,h_0]$, define the semiclassical Kohn-Nirenberg quantization $\operatorname{Op}_h(a): \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)$ by the oscillatory integral
for $u \in \mathcal{S}(\mathbb{R}^n)$ and $x \in \mathbb{R}^n$. Then there exist an integer $N=N(n)\in \mathbb{N}$ and a constant $C=C(n,h_0)>0$ such that, for every $h \in (0,h_0]$ and every $u \in \mathcal{S}(\mathbb{R}^n)$,