Fix $n\in\mathbb{N}$ and $h_0>0$. For $r\in\mathbb{R}$, let $H_h^r(\mathbb{R}^n)$ denote the semiclassical Sobolev space with norm \begin{align*} \|u\|_{H_h^r}=\|\langle hD\rangle^r u\|_{L^2(\mathbb{R}^n)} \end{align*} where $\langle hD\rangle^r$ is the Fourier multiplier with symbol $\langle h\xi\rangle^r=(1+h^2|\xi|^2)^{r/2}$. Let $(R_h)_{0<h\leq h_0}$ be a family of continuous operators $C_c^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^n)$ with smooth Schwartz kernels $K_h:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{C}$. Assume that for every pair of multi-indices $\alpha,\beta\in\mathbb{N}_0^n$ and every $L,N\in\mathbb{N}$ there is a constant $C_{\alpha,\beta,L,N}>0$ such that \begin{align*} \sup_{x,y\in\mathbb{R}^n}(1+|x-y|)^L|\partial_x^\alpha\partial_y^\beta K_h(x,y)|\leq C_{\alpha,\beta,L,N}h^N \end{align*} for all $0<h\leq h_0$. Then for every $s,t\in\mathbb{R}$ and every $N\in\mathbb{N}$ there is a constant $C_{s,t,N}>0$ such that \begin{align*} \|R_h u\|_{H_h^t}\leq C_{s,t,N}h^N\|u\|_{H_h^{-s}} \end{align*} for all $u\in C_c^\infty(\mathbb{R}^n)$ and all $0<h\leq h_0$. Conversely, suppose that for every $\chi,\psi\in C_c^\infty(\mathbb{R}^n)$, every $s,t\in\mathbb{R}$, and every $N\in\mathbb{N}$, the localized family $\chi R_h\psi$ satisfies \begin{align*} \|\chi R_h\psi u\|_{H_h^t}\leq C_{\chi,\psi,s,t,N}h^N\|u\|_{H_h^{-s}} \end{align*} for all $u\in C_c^\infty(\mathbb{R}^n)$ and all $0<h\leq h_0$. Then for every compact set $E\subset\mathbb{R}^n\times\mathbb{R}^n$, every pair of multi-indices $\alpha,\beta\in\mathbb{N}_0^n$, and every $N\in\mathbb{N}$ there is a constant $C_{E,\alpha,\beta,N}>0$ such that \begin{align*} \sup_{(x,y)\in E}|\partial_x^\alpha\partial_y^\beta K_h(x,y)|\leq C_{E,\alpha,\beta,N}h^N \end{align*} for all $0<h\leq h_0$.