Let $X$ be a smooth compact manifold, let $h_0 > 0$, let $m,s \in \mathbb{R}$, and let $A=(A_h)_{0<h\le h_0} \in \Psi_h^m(X)$ be a semiclassical pseudodifferential operator with semiclassical principal symbol $\sigma_{h,m}(A):T^*X \to \mathbb{C}$. Let $U \subset T^*X$ be open, and assume that $A$ is elliptic on $U$, meaning $\sigma_{h,m}(A)(\rho) \neq 0$ for every $\rho \in U$.

Let $u = (u_h)_{0<h\le h_0}$ be a family of distributions $u_h \in \mathcal{D}'(X)$ that is semiclassically tempered in the following sense: there exists $N_0 \in \mathbb{N}$ such that $u_h \in H_h^{-N_0}(X)$ for all sufficiently small $h$ and $\|u_h\|_{H_h^{-N_0}(X)} = O(h^{-N_0})$ as $h \to 0$.

For an operator $C=(C_h)_{0<h\le h_0} \in \Psi_h^0(X)$, write $\operatorname{WF}_h'(C) \subset T^*X$ for its semiclassical operator wavefront set, also called its compact semiclassical microsupport when it is compact. The expression $Au$ denotes the family $(A_hu_h)_{0<h\le h_0}$. Assume that $Au$ is microlocally bounded in $H_h^{s-m}$ on $U$: for every $C_1=(C_{1,h})_{0<h\le h_0} \in \Psi_h^0(X)$ with $\operatorname{WF}_h'(C_1)$ compact and contained in $U$, $C_{1,h}A_hu_h \in H_h^{s-m}(X)$ for all sufficiently small $h$ and $\|C_1 A u_h\|_{H_h^{s-m}(X)} = O(1)$.

Then $u$ is microlocally bounded in $H_h^s$ on $U$: for every $C=(C_h)_{0<h\le h_0} \in \Psi_h^0(X)$ with $\operatorname{WF}_h'(C)$ compact and contained in $U$, $C_hu_h \in H_h^s(X)$ for all sufficiently small $h$ and $\|C u_h\|_{H_h^s(X)} = O(1)$.