Let $X$ be a smooth manifold equipped with a fixed smooth positive density, and let $P:C_c^\infty(X)\to C^\infty(X)$ be a properly supported classical pseudodifferential operator of order $m$, extended continuously to $P:\mathcal D'(X)\to \mathcal D'(X)$. Assume that the scalar principal symbol of $P$ has a real homogeneous representative $p\in C^\infty(T^*X\setminus 0;\mathbb R)$. Assume also that the chosen quantization satisfies the standard pseudodifferential adjoint and composition formulae.

Fix $s\in\mathbb R$. Let $a\in S^{s-(m-1)/2}(T^*X\setminus 0)$ be real-valued, homogeneous for large fibre norm, and supported in a compact conic subset of a coordinate patch after quotienting by positive fibre rescaling. Let $A\in \Psi^{s-(m-1)/2}(X)$ be a properly supported quantization of $a$.

Suppose that there exist real symbols $b,e,g\in S^s(T^*X\setminus 0)$ and $r\in S^{2s-1}(T^*X\setminus 0)$ such that $g$ is elliptic on a conic neighbourhood of $\operatorname{supp} r$, and, microlocally near $\operatorname{WF}'(A)$, $\frac{1}{i}\{p,a^2\}=b^2-e^2+r$. Let $B,E,G\in \Psi^s(X)$ and $R\in \Psi^{2s-1}(X)$ be properly supported quantizations of $b,e,g,r$, with $\operatorname{WF}'(R)$ contained in the elliptic set of $G$.

Assume that the sign construction and the chosen cutoffs cover $\operatorname{WF}'(A)$ in the following explicit sense: for every properly supported cutoff $Q_0\in\Psi^0(X)$ with $Q_0=1$ on $\operatorname{WF}'(A)$ in the conic region under consideration, there is a properly supported cutoff $Q_1\in\Psi^0(X)$ with $Q_1=1$ on $\operatorname{WF}'(Q_0)$, the microsupports of $1-Q_0$ and $A$ are disjoint in the fixed conic region, the microsupports of $1-Q_1$ and $Q_0$ are disjoint there, and a constant $C>0$ such that, uniformly for the regularized operators below, $\|A_\varepsilon Q_0 v\|_{H^{(m-1)/2}}\le C(\|B_\varepsilon v\|_{L^2}+\|E_\varepsilon v\|_{L^2}+\|Gv\|_{L^2}+\|v\|_{H^{s-1/2}})$ for all $v\in C_c^\infty(X)$ localized in the coordinate patch. Equivalently, the elliptic sets of $B$, $E$, and $G$ generated by the sign construction cover the part of $\operatorname{WF}'(A)$ needed to control $A$ at order $s-(m-1)/2$.

Assume that, after shrinking all conic cutoffs around $\operatorname{WF}'(A)$, the real-principal-symbol part of the commutator admits the microlocal decomposition $\frac{1}{i}[P,A^*A]=B^*B-E^*E+R+F+\mathcal K$, where $F\in \Psi^{2s-2}(X)$ and $\mathcal K$ denotes the separated energy contribution of the order $m-1$ skew-adjoint part of $P$.

For the regularized commutants $A_\varepsilon=A\Lambda_\varepsilon$, $0<\varepsilon\le 1$, assume that $\Lambda_\varepsilon\in\Psi^0(X)$ is uniformly bounded on all Sobolev spaces under consideration, smoothing for each $\varepsilon>0$, and converges microlocally to the identity as $\varepsilon\to 0$. Assume that the corresponding operators $B_\varepsilon,E_\varepsilon,R_\varepsilon,F_\varepsilon,\mathcal K_\varepsilon$ satisfy the same microlocal decomposition with symbol seminorms uniformly bounded in $\varepsilon$, that $B_\varepsilon\to B$ distributionally as operators on localized distributions, and that $E_\varepsilon$ is compatible with $E$ in the sense that $\|E_\varepsilon v\|_{L^2}\le C(\|Ev\|_{L^2}+\|v\|_{H^{s-1/2}})$ for all localized $v\in C_c^\infty(X)$, with $C$ independent of $\varepsilon$.

Finally assume the following subprincipal absorption condition. For every such regularized commutant, the energy contribution of the order $m-1$ skew-adjoint part of $P$ is bounded, for every $\delta>0$, by $\delta\|B_\varepsilon u\|_{L^2}^2+C_\delta(\|E_\varepsilon u\|_{L^2}^2+\|Gu\|_{L^2}^2+\|u\|_{H^{s-1/2}}^2)$, with $C_\delta$ independent of $\varepsilon$ and $u$. This condition is automatic in the microlocally self-adjoint model case $P-P^*\in\Psi^{m-2}(X)$.

Let $u\in\mathcal D'(X)$ be compactly supported in the coordinate patch, or first localized there by a properly supported cutoff. Choose fixed properly supported microlocal cutoffs equal to $1$ on the displayed microsupports, and use them to define the microlocal Sobolev norm $\|Pu\|_{H^{s-m+1}(\operatorname{WF}'(A))}$. Assume that $u$ has the localized lower regularity measured by the displayed $H^{s-1/2}$ remainder. Assume that the localization and smoothing family are chosen so that every localized distribution satisfying the hypotheses admits smooth localized approximants $u_j\in C_c^\infty(X)$ with $u_j\to u$ in $\mathcal D'(X)$, $Eu_j\to Eu$ and $Gu_j\to Gu$ in local $L^2$, $Q_1Pu_j\to Q_1Pu$ in $H^{s-m+1}(X)$ for the fixed cutoff $Q_1$ defining the displayed microlocal norm, and $u_j\to u$ in the localized $H^{s-1/2}$ norm appearing in the estimate.

If $Gu\in L^2_{\mathrm{loc}}(X)$, $Eu\in L^2_{\mathrm{loc}}(X)$, and $Pu$ is microlocally in $H^{s-m+1}$ on a conic neighbourhood of $\operatorname{WF}'(A)$, then $Bu\in L^2_{\mathrm{loc}}(X)$. More precisely, there exists a constant $C>0$, depending only on the fixed cutoffs, the cutoff separation constants, the elliptic parametrix bounds, the symbol seminorms in the chosen coordinate patch, and the absorption constants, but independent of $\varepsilon$, such that $\|Bu\|_{L^2}^2 \le C(\|Eu\|_{L^2}^2+\|Gu\|_{L^2}^2+\|Pu\|_{H^{s-m+1}(\operatorname{WF}'(A))}^2+\|u\|_{H^{s-1/2}}^2)$ after inserting the fixed cutoffs equal to $1$ on the displayed microsupports.