[proofplan]
The implication from rapid $C^\infty$ decay to empty wavefront set follows directly from the definition: every order-zero semiclassical pseudodifferential operator sends an $O(h^\infty)_{C^\infty}$ family to another such family. Conversely, if the compactified wavefront set is empty, every point of the compact space $\overline{T}^*X$ has an elliptic microlocal cutoff that annihilates $u_h$ rapidly. Compactness gives finitely many such cutoffs, and a microlocal partition of unity converts them into an identity formula modulo a residual $O(h^\infty)$ operator. Applying that identity to the tempered family $u_h$ gives rapid decay in $C^\infty(X)$.
[/proofplan]
[step:Show that rapid smooth decay excludes every compactified covector]
Assume $u_h=O(h^\infty)_{C^\infty(X)}$. Let $q \in \overline{T}^*X$ be arbitrary. To show $q\notin \operatorname{WF}_h(u_h)$, let $A_h\in\Psi_h^0(X)$ be any properly supported order-zero compactified semiclassical pseudodifferential operator, viewed as a continuous map $A_h:\mathcal{D}'(X)\to\mathcal{D}'(X)$, whose compactified semiclassical principal symbol is elliptic at $q$.
For every continuous seminorm $p$ on $C^\infty(X)$, continuity of order-zero semiclassical pseudodifferential operators on smooth functions gives a continuous seminorm $p_A$ on $C^\infty(X)$ and an integer $M_A\geq 0$ such that
\begin{align*}
p(A_h v) \leq C_A h^{-M_A} p_A(v)
\end{align*}
for all $v\in C^\infty(X)$ and all sufficiently small $h$. Applying this with $v=u_h$ and using $u_h=O(h^\infty)_{C^\infty(X)}$, we obtain
\begin{align*}
p(A_h u_h) = O(h^\infty).
\end{align*}
Thus $A_h u_h=O(h^\infty)_{C^\infty(X)}$. Since $A_h$ was elliptic at the arbitrary point $q$, the definition of the compactified semiclassical wavefront set gives $q\notin\operatorname{WF}_h(u_h)$. Hence $\operatorname{WF}_h(u_h)=\varnothing$.
[guided]
Assume $u_h=O(h^\infty)_{C^\infty(X)}$. We want to prove that no point $q\in\overline{T}^*X$ belongs to the compactified semiclassical wavefront set. By definition, it is enough to show that whenever an order-zero semiclassical pseudodifferential operator is elliptic at $q$, applying it to $u_h$ still produces an $O(h^\infty)$ smooth family.
Fix an arbitrary compactified covector $q\in\overline{T}^*X$, and let $A_h\in\Psi_h^0(X)$ be a properly supported order-zero compactified semiclassical pseudodifferential operator, viewed as a continuous map $A_h:\mathcal{D}'(X)\to\mathcal{D}'(X)$, elliptic at $q$. Order-zero semiclassical operators act continuously on smooth functions, with at most a fixed polynomial loss in $h$. Therefore, for every continuous seminorm $p$ on $C^\infty(X)$, there are a continuous seminorm $p_A$ on $C^\infty(X)$, an integer $M_A\geq 0$, and a constant $C_A>0$ such that
\begin{align*}
p(A_h v) \leq C_A h^{-M_A}p_A(v)
\end{align*}
for all $v\in C^\infty(X)$ and all sufficiently small $h$.
Now use the rapid decay hypothesis. Since $u_h=O(h^\infty)_{C^\infty(X)}$, for every integer $N\in\mathbb N$ we may bound $p_A(u_h)$ by a constant times $h^{N+M_A}$. Substituting this into the operator estimate gives
\begin{align*}
p(A_h u_h) \leq C_A h^{-M_A} p_A(u_h) = O(h^N).
\end{align*}
Because $N$ was arbitrary, $p(A_hu_h)=O(h^\infty)$ for every smooth seminorm $p$. Hence $A_hu_h=O(h^\infty)_{C^\infty(X)}$.
This proves that the elliptic test at the point $q$ succeeds, so $q\notin\operatorname{WF}_h(u_h)$. Since $q$ was arbitrary in $\overline{T}^*X$, the compactified semiclassical wavefront set is empty.
[/guided]
[/step]
[step:Choose finitely many elliptic cutoffs annihilating $u_h$ rapidly]
Assume now that $\operatorname{WF}_h(u_h)=\varnothing$. For each point $q\in\overline{T}^*X$, the definition of the complement of the wavefront set gives an open neighbourhood $U_q\subset\overline{T}^*X$ of $q$ and an operator $A_{q,h}\in\Psi_h^0(X)$, viewed as a continuous map $A_{q,h}:\mathcal{D}'(X)\to\mathcal{D}'(X)$, that is elliptic on $U_q$ and satisfies
\begin{align*}
A_{q,h}u_h=O(h^\infty)_{C^\infty(X)}.
\end{align*}
Since $X$ is compact, the radial compactification $\overline{T}^*X$ is compact. Choose finitely many points $q_1,\dots,q_J\in\overline{T}^*X$ such that the corresponding elliptic neighbourhoods $U_{q_1},\dots,U_{q_J}$ cover $\overline{T}^*X$. Write
\begin{align*}
A_{j,h}:=A_{q_j,h}
\end{align*}
for $1\leq j\leq J$. Then each $A_{j,h}\in\Psi_h^0(X)$ is elliptic on $U_{q_j}$ and
\begin{align*}
A_{j,h}u_h=O(h^\infty)_{C^\infty(X)}
\end{align*}
for every $1\leq j\leq J$.
[/step]
[step:Build a finite microlocal partition of unity subordinate to the elliptic cover]
Let $a_j\in S^0(\overline{T}^*X)$ denote the compactified principal symbol of $A_{j,h}$, where $S^0(\overline{T}^*X)$ is the order-zero compactified semiclassical symbol class. Since $A_{j,h}$ is elliptic on $U_{q_j}$, the finite elliptic parametrix construction for a cover of $\overline{T}^*X$ gives full symbols $b_j\in S^0(\overline{T}^*X)$ with compactified microsupport contained in $U_{q_j}$ such that, after quantization, the full symbolic error in $\sum_{j=1}^J B_{j,h}A_{j,h}-I$ lies in $h^\infty S^{-\infty}$, the residual compactified semiclassical symbol class with all symbol seminorms $O(h^\infty)$. Let $B_{j,h}\in\Psi_h^0(X)$ be an order-zero compactified semiclassical pseudodifferential operator with full symbol $b_j$, viewed on smooth functions as a map $B_{j,h}:C^\infty(X)\to C^\infty(X)$.
The semiclassical composition calculus and the full-symbol parametrix construction give an operator identity
\begin{align*}
I = \sum_{j=1}^J B_{j,h}A_{j,h} + R_h
\end{align*}
on distributions, where $R_h:\mathcal{D}'(X)\to C^\infty(X)$ is a residual semiclassical smoothing operator whose full symbol is $O(h^\infty)$ in $S^{-\infty}$.
[guided]
The purpose of this step is to turn local microlocal information into a global identity. We have finitely many elliptic operators $A_{j,h}$, and the ellipticity of $A_{j,h}$ on $U_{q_j}$ means that its principal symbol $a_j$ is invertible there. Because the open sets $U_{q_j}$ cover the whole compactified cotangent bundle $\overline{T}^*X$, we can choose symbols $b_j$ supported microlocally inside $U_{q_j}$ so that the products $b_j a_j$ add up to $1$ microlocally.
More precisely, let $a_j\in S^0(\overline{T}^*X)$ be the compactified principal symbol of $A_{j,h}$, where $S^0(\overline{T}^*X)$ denotes the order-zero compactified semiclassical symbol class. On $U_{q_j}$, the symbol $a_j$ is elliptic, so one may divide by $a_j$ there at the symbolic level. Choosing a smooth partition of unity on the compact space $\overline{T}^*X$ subordinate to the finite cover $U_{q_1},\dots,U_{q_J}$ gives a principal-symbol inverse. To upgrade this principal inverse to an $O(h^\infty)$ inverse, one uses the full semiclassical composition formula recursively: after the principal term has killed the order-one error, the next coefficient in the full symbol of each $b_j$ is chosen to kill the next power of $h$, and continuing this construction gives a full symbol whose remaining error lies in $h^\infty S^{-\infty}$. Here $S^{-\infty}$ is the residual compactified semiclassical symbol class, and $h^\infty S^{-\infty}$ means that every residual symbol seminorm is $O(h^N)$ for every $N\in\mathbb{N}$.
Now quantize each full symbol $b_j$ to an order-zero semiclassical pseudodifferential operator $B_{j,h}:C^\infty(X)\to C^\infty(X)$. The semiclassical composition calculus says that the full symbol of $\sum_{j=1}^J B_{j,h}A_{j,h}$ equals the full symbol of the identity modulo $h^\infty S^{-\infty}$. Therefore the sum of the compositions recovers the identity modulo a residual smoothing operator:
\begin{align*}
I = \sum_{j=1}^J B_{j,h}A_{j,h} + R_h.
\end{align*}
Here $R_h:\mathcal{D}'(X)\to C^\infty(X)$ is residual, meaning its Schwartz kernel is smooth and all of its symbol seminorms are $O(h^\infty)$. This is the point where compactness is essential: without a finite cover, we would not get a finite operator identity of this form.
[/guided]
[/step]
[step:Apply the parametrix identity to the tempered family]
Apply the identity from the previous step to $u_h$:
\begin{align*}
u_h = \sum_{j=1}^J B_{j,h}A_{j,h}u_h + R_hu_h.
\end{align*}
For each $j$, the family $A_{j,h}u_h$ is $O(h^\infty)_{C^\infty(X)}$, and $B_{j,h}\in\Psi_h^0(X)$ maps rapidly decaying smooth families to rapidly decaying smooth families. Hence
\begin{align*}
B_{j,h}A_{j,h}u_h=O(h^\infty)_{C^\infty(X)}
\end{align*}
for every $1\leq j\leq J$.
It remains to control $R_hu_h$. By semiclassical temperedness as stated in the theorem, choose $s_0\in\mathbb R$, $M_0\geq0$, $C_0>0$, and $h_1\in(0,h_0]$ such that $\|u_h\|_{H_h^{-s_0}(X)}\leq C_0h^{-M_0}$ for $0<h\leq h_1$, where $H_h^{-s_0}(X)$ is the semiclassical Sobolev space of order $-s_0$. Since $R_h$ is residual with $O(h^\infty)$ smoothing seminorms, for every continuous seminorm $p$ on $C^\infty(X)$ and every $N\in\mathbb N$ there are constants $C_{p,N}>0$ and $h_{p,N}\in(0,h_1]$ such that
\begin{align*}
p(R_h v) \leq C_{p,N}h^{N+M_0}\|v\|_{H_h^{-s_0}(X)}
\end{align*}
for all distributions $v\in H_h^{-s_0}(X)$ and all $0<h\leq h_{p,N}$. Applying this estimate to $v=u_h$ and using the temperedness bound gives
\begin{align*}
p(R_hu_h)=O(h^N).
\end{align*}
Since $N$ and $p$ were arbitrary, $R_hu_h=O(h^\infty)_{C^\infty(X)}$.
The right-hand side of the identity for $u_h$ is a finite sum of $O(h^\infty)_{C^\infty(X)}$ families. Hence
\begin{align*}
u_h=O(h^\infty)_{C^\infty(X)}.
\end{align*}
This proves the converse implication and completes the proof.
[/step]
