Let $n \in \mathbb{N}$, let $h \in (0,h_0]$ with $h_0 > 0$, let $m \in \mathbb{R}$, and let $a \in S^m(T^*\mathbb{R}^n)$ be a semiclassical symbol. Define the semiclassical left quantization

\begin{align*} \operatorname{Op}_h(a)u(x) := \frac{1}{(2\pi h)^n}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} e^{i(x-y)\cdot \xi/h} a(x,\xi;h)u(y)\, d\mathcal{L}^n(y)\, d\mathcal{L}^n(\xi) \end{align*}

for $u \in \mathcal{S}(\mathbb{R}^n)$, with the $\xi$-integral interpreted as an oscillatory integral. Then $\operatorname{Op}_h(a): \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)$ has Schwartz kernel $K_a \in \mathcal{S}'(\mathbb{R}^n_x \times \mathbb{R}^n_y)$ given by

\begin{align*} K_a(x,y;h) = \frac{1}{(2\pi h)^n}\int_{\mathbb{R}^n} e^{i(x-y)\cdot \xi/h}a(x,\xi;h)\, d\mathcal{L}^n(\xi) \end{align*}

in the sense of oscillatory integrals. Equivalently, for every $u,v \in \mathcal{S}(\mathbb{R}^n)$,

\begin{align*} \operatorname{Op}_h(a)u(v) = K_a(v \otimes u). \end{align*}

Moreover, $K_a$ is represented by a smooth function on $\{(x,y) \in \mathbb{R}^n \times \mathbb{R}^n : x \ne y\}$. For every compact set $E \subset \{(x,y) : x \ne y\}$, every multiindices $\alpha,\beta \in \mathbb{N}_0^n$, and every $N \in \mathbb{N}$ with $N > m + |\alpha|$, there is a constant $C_{\alpha,\beta,N,E} > 0$ such that

\begin{align*} |\partial_x^\alpha \partial_y^\beta K_a(x,y;h)| \le C_{\alpha,\beta,N,E} h^{N-n-|\alpha|-|\beta|} \end{align*}

for all $(x,y) \in E$ and all $h \in (0,h_0]$.