[proofplan]
We prove the result by applying the semiclassical elliptic estimate locally near each point of $U$. Given a cutoff $C$ supported inside the elliptic set of $A$, we choose a second cutoff $C_1$ that equals the identity microlocally on the microsupport of $C$. The elliptic estimate bounds $\|C u_h\|_{H_h^s}$ by the known quantity $\|C_1 A u_h\|_{H_h^{s-m}}$ plus a smoothing remainder. The assumed polynomial bound on $u_h$ absorbs that remainder after choosing the smoothing order sufficiently large.
[/proofplan]
[step:Localize the estimate to an elliptic neighborhood inside $U$]
Fix an operator $C \in \Psi_h^0(X)$ such that its semiclassical operator wavefront set $\operatorname{WF}_h'(C) \subset T^*X$, also called its semiclassical microsupport, is compact and contained in $U$. Since $A$ is elliptic on $U$, it is elliptic on an open neighborhood of the compact set $\operatorname{WF}_h'(C)$.
Choose $C_1 \in \Psi_h^0(X)$ with $\operatorname{WF}_h'(C_1)$ compact and contained in $U$ such that $C_1$ is microlocally equal to the identity on a neighborhood of $\operatorname{WF}_h'(C)$. These are exactly the parametrix hypotheses for the standard semiclassical microlocal elliptic estimate: $A$ is elliptic on a neighborhood of $\operatorname{WF}_h'(C)$, and $C_1$ is microlocally equal to the identity on that neighborhood. Applying that estimate to $A$ and to the cutoffs $C,C_1$, for every $N \in \mathbb{N}$ there exists a constant $K_N > 0$ such that
\begin{align*}
\|C u_h\|_{H_h^s(X)} \le K_N \|C_1 A u_h\|_{H_h^{s-m}(X)} + K_N h^N \|u_h\|_{H_h^{-N}(X)}
\end{align*}
for all sufficiently small $h$.
Here $K_N$ depends on $A,C,C_1,s,m,N$ but not on $h$ or on $u_h$.
[guided]
We want to measure $u_h$ only near the compact microlocal region where $C$ sees it. The notation $\operatorname{WF}_h'(C)$ denotes the semiclassical operator wavefront set of $C$, namely the microlocal support of the Schwartz kernel of $C$ viewed as an operator on $X$. Because $\operatorname{WF}_h'(C)$ is compact and contained in the open set $U$, and because $A$ is elliptic at every point of $U$, the operator $A$ is elliptic on an open neighborhood of $\operatorname{WF}_h'(C)$.
The auxiliary cutoff $C_1 \in \Psi_h^0(X)$ is chosen so that $\operatorname{WF}_h'(C_1)$ remains compact and contained in $U$ and so that $C_1$ is microlocally the identity near $\operatorname{WF}_h'(C)$. These two facts are the parametrix input needed for the standard semiclassical microlocal elliptic estimate: the elliptic inverse is constructed on the region where $C$ localizes, while $C_1$ localizes the resulting application of $A u_h$ to the same elliptic region.
The semiclassical microlocal elliptic estimate then gives, for every chosen smoothing order $N \in \mathbb{N}$, a constant $K_N > 0$ with
\begin{align*}
\|C u_h\|_{H_h^s(X)} \le K_N \|C_1 A u_h\|_{H_h^{s-m}(X)} + K_N h^N \|u_h\|_{H_h^{-N}(X)}.
\end{align*}
The first term is the elliptic recovery term: applying $A$ costs $m$ derivatives, so $A u_h$ is measured in $H_h^{s-m}$. The second term is the smoothing remainder from the microlocal parametrix. It is multiplied by an arbitrary power $h^N$, which is why the temperedness assumption on $u_h$ will be enough to control it.
[/guided]
[/step]
[step:Use the hypothesis on $Au$ to bound the principal term]
The operator $C_1$ has compact semiclassical microsupport contained in $U$. Therefore the microlocal $H_h^{s-m}$ hypothesis on $Au$ gives
\begin{align*}
\|C_1 A u_h\|_{H_h^{s-m}(X)} = O(1)
\end{align*}
as $h \to 0$.
[/step]
[step:Choose the smoothing order to absorb the polynomial growth of $u$]
By semiclassical temperedness, there exists $N_0 \in \mathbb{N}$ such that
\begin{align*}
\|u_h\|_{H_h^{-N_0}(X)} = O(h^{-N_0}).
\end{align*}
Choose an integer $N > N_0$. Since semiclassical Sobolev norms become weaker as the order decreases, increasing $N$ if necessary gives
\begin{align*}
\|u_h\|_{H_h^{-N}(X)} \le M_N \|u_h\|_{H_h^{-N_0}(X)}
\end{align*}
for some constant $M_N > 0$ independent of $h$. Hence
\begin{align*}
h^N \|u_h\|_{H_h^{-N}(X)} = O(h^{N-N_0}).
\end{align*}
Because $N-N_0>0$, this remainder is $O(1)$, and in fact tends to $0$ as $h \to 0$.
[guided]
The elliptic estimate contains a remainder term of the form
\begin{align*}
h^N \|u_h\|_{H_h^{-N}(X)}.
\end{align*}
The point of the temperedness hypothesis is to control this term even though $u_h$ need not be uniformly bounded in any fixed Sobolev norm. By assumption, there is an integer $N_0 \in \mathbb{N}$ such that
\begin{align*}
\|u_h\|_{H_h^{-N_0}(X)} = O(h^{-N_0}).
\end{align*}
We choose an integer $N > N_0$. The Sobolev space $H_h^{-N_0}(X)$ continuously embeds into the weaker space $H_h^{-N}(X)$, so there is a constant $M_N > 0$, independent of $h$, such that
\begin{align*}
\|u_h\|_{H_h^{-N}(X)} \le M_N \|u_h\|_{H_h^{-N_0}(X)}.
\end{align*}
Multiplying by $h^N$ gives
\begin{align*}
h^N \|u_h\|_{H_h^{-N}(X)} = O(h^{N-N_0}).
\end{align*}
Since $N-N_0>0$, the right-hand side is bounded as $h \to 0$ and even tends to $0$. Thus the smoothing remainder in the elliptic estimate is harmless.
[/guided]
[/step]
[step:Conclude the microlocal $H_h^s$ bound on $U$]
Substituting the two bounds into the elliptic estimate gives
\begin{align*}
\|C u_h\|_{H_h^s(X)} \le O(1) + O(1) = O(1).
\end{align*}
Since $C \in \Psi_h^0(X)$ was arbitrary with $\operatorname{WF}_h'(C)$ compact and contained in $U$, this proves that $u$ is microlocally bounded in $H_h^s$ on $U$.
[guided]
We now return to the elliptic estimate for the fixed cutoff $C$:
\begin{align*}
\|C u_h\|_{H_h^s(X)} \le K_N \|C_1 A u_h\|_{H_h^{s-m}(X)} + K_N h^N \|u_h\|_{H_h^{-N}(X)}.
\end{align*}
The first term is $O(1)$ by the microlocal $H_h^{s-m}$ hypothesis on $Au$, because $\operatorname{WF}_h'(C_1)$ is compact and contained in $U$. The second term is $O(1)$ by the previous smoothing-remainder estimate. Therefore
\begin{align*}
\|C u_h\|_{H_h^s(X)} = O(1).
\end{align*}
The cutoff $C \in \Psi_h^0(X)$ was arbitrary subject only to the condition that $\operatorname{WF}_h'(C)$ is compact and contained in $U$. This is precisely the definition of $u$ being microlocally bounded in $H_h^s$ on $U$.
[/guided]
[/step]
