Let $X$ be a smooth manifold equipped with the scalar semiclassical pseudodifferential calculus, let $m,s\in\mathbb{R}$, and let $A=(A_h)_{0<h<h_*}\in\Psi_h^m(X)$ be a scalar semiclassical pseudodifferential operator family on compactly supported scalar distributions. Let $a_m$ denote the scalar semiclassical principal symbol of $A$, and assume that $a_m$ is semiclassically elliptic at a point $(x_0,\xi_0)\in T^*X$. Here $\Psi_h^r(X)$ denotes the order $r$ scalar semiclassical pseudodifferential operators on $X$, $\operatorname{WF}_h(P)\subset T^*X$ denotes the semiclassical operator wavefront set of $P\in\Psi_h^r(X)$, and $H_h^t(X)$ denotes the semiclassical Sobolev space of order $t\in\mathbb{R}$. Then there exists an open neighbourhood $V\subset T^*X$ of $(x_0,\xi_0)$ with the following property.

For every operator family $C=(C_h)_{0<h<h_*}\in\Psi_h^0(X)$ satisfying $\operatorname{WF}_h(C)\Subset V$, there exist operator families $B=(B_h)_{0<h<h_*}\in\Psi_h^{-m}(X)$ and $C_1=(C_{1,h})_{0<h<h_*}\in\Psi_h^0(X)$, with $C_1$ elliptic on $\operatorname{WF}_h(C)$, and a number $h_0>0$ independent of $N$ such that, for every $N\in\mathbb{N}$, there is a constant $C_N>0$ for which

\begin{align*} \|Cu\|_{H_h^s(X)} \leq C_N\|C_1Au\|_{H_h^{s-m}(X)} + C_N h^N\|u\|_{H_h^{-N}(X)} \end{align*}

for all $0<h<h_0$ and all $h$-dependent compactly supported scalar distributions $u=(u_h)_{0<h<h_0}$ with $u_h\in\mathcal E'(X)$ for which the two norms on the right-hand side are finite.