Let $n \in \mathbb{N}$, let $h_0 > 0$, and let $(u_h)_{0<h\leq h_0}$ be a family in $\mathcal{S}(\mathbb{R}^n)$. Assume the family is uniformly Schwartz, meaning that for every pair of multi-indices $\alpha,\gamma \in \mathbb{N}_0^n$ there exists $M_{\alpha\gamma} > 0$ such that

\begin{align*} \sup_{0<h\leq h_0}\sup_{x\in\mathbb{R}^n}|x^\alpha D_x^\gamma u_h(x)|\leq M_{\alpha\gamma}. \end{align*}

Here $D_{x_j}:=-i\partial_{x_j}$ and $D_x^\gamma:=D_{x_1}^{\gamma_1}\cdots D_{x_n}^{\gamma_n}$.

Then for every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$ there exists $C_{\alpha\beta}>0$ such that

\begin{align*} \sup_{0<h\leq h_0}\sup_{x\in\mathbb{R}^n}|x^\alpha (hD_x)^\beta u_h(x)|\leq C_{\alpha\beta}, \end{align*}

where $(hD_x)^\beta:=\prod_{j=1}^n(hD_{x_j})^{\beta_j}$.

Moreover, let $a \in \mathcal{S}(\mathbb{R}^n)$ be fixed, and define the ordinary Fourier transform $\mathcal{F}a:\mathbb{R}^n\to\mathbb{C}$ by

\begin{align*} \mathcal{F}a(\eta)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}a(x)e^{-ix\cdot\eta}\,d\mathcal{L}^n(x). \end{align*}

For the ordinary Fourier variable $\eta\in\mathbb{R}^n$, set $D_{\eta_j}:=-i\partial_{\eta_j}$ and $D_\eta^\beta:=D_{\eta_1}^{\beta_1}\cdots D_{\eta_n}^{\beta_n}$. For the semiclassical Fourier variable $\xi\in\mathbb{R}^n$, set $D_{\xi_j}:=-i\partial_{\xi_j}$ and $D_\xi^\beta:=D_{\xi_1}^{\beta_1}\cdots D_{\xi_n}^{\beta_n}$. For $0<h\leq h_0$, define the semiclassical Fourier transform $\mathcal{F}_h a:\mathbb{R}^n\to\mathbb{C}$ by

\begin{align*} \mathcal{F}_h a(\xi)=h^{-n/2}\mathcal{F}a(\xi/h). \end{align*}

Then for every integer $N\geq 0$ and every multi-index $\beta\in\mathbb{N}_0^n$ there exists $C_{N\beta}>0$ such that

\begin{align*} \sup_{0<h\leq h_0}\sup_{\xi\in\mathbb{R}^n}h^{n/2+|\beta|}\left(1+\frac{|\xi|}{h}\right)^N|D_\xi^\beta\mathcal{F}_h a(\xi)|\leq C_{N\beta}. \end{align*}