Let $X$ be a compact smooth manifold without boundary, let $h_0>0$, and let $(u_h)_{0<h\leq h_0}$ be a family in $\mathcal{D}'(X)$. Assume $(u_h)$ is semiclassically tempered, meaning that there are $s_0\in\mathbb{R}$, $M_0\geq 0$, $C_0>0$, and $h_1\in(0,h_0]$ such that $\|u_h\|_{H_h^{-s_0}(X)}\leq C_0 h^{-M_0}$ for all $0<h\leq h_1$, where $H_h^{-s_0}(X)$ denotes the semiclassical Sobolev space of order $-s_0$ on $X$. Then the compactified semiclassical wavefront set of $u_h$ is empty in the radial compactification $\overline{T}^*X$ if and only if $u_h$ is rapidly decaying in every smooth seminorm:

\begin{align*} \operatorname{WF}_h(u_h)=\varnothing \iff u_h = O(h^\infty)_{C^\infty(X)}. \end{align*}

Equivalently, for every continuous seminorm $p:C^\infty(X)\to[0,\infty)$ and every $N\in\mathbb{N}$, there are constants $C_{p,N}>0$ and $h_{p,N}\in(0,h_0]$ such that $p(u_h)\leq C_{p,N}h^N$ for all $0<h\leq h_{p,N}$.