[proofplan] We first prove the estimate for smooth compactly supported $u$ and regularized commutants $A_\varepsilon=A\Lambda_\varepsilon$, so that every pairing below is an ordinary $L^2$ pairing. The commutator identity expresses the positive term $\|B_\varepsilon u\|_{L^2}^2$ in terms of the incoming term, the elliptic error, the lower-order remainder, the $Pu$ pairing, and the separated skew-adjoint contribution. The skew-adjoint contribution is absorbed by hypothesis, the $Pu$ pairing is controlled by placing one $A_\varepsilon$ on each factor and using Sobolev duality plus microlocal elliptic control, and the $R_\varepsilon$ and $F_\varepsilon$ terms are controlled by $G$ and the lower Sobolev norm. The resulting estimate is uniform in $\varepsilon$; weak compactness and convergence of the regularized operators then give the unregularized estimate, and localization plus density extend it to distributions. [/proofplan] [step:Regularize the commutant and record the uniform symbolic decomposition] Let \begin{align*} \Lambda_\varepsilon\in\Psi^0(X),\qquad 0<\varepsilon\le 1, \end{align*} be the regularizing family from the statement, uniformly bounded on the Sobolev spaces used below and smoothing for each fixed $\varepsilon>0$. Define \begin{align*} A_\varepsilon:=A\Lambda_\varepsilon\in\Psi^{s-(m-1)/2}(X). \end{align*} Let \begin{align*} B_\varepsilon,E_\varepsilon\in\Psi^s(X),\qquad R_\varepsilon\in\Psi^{2s-1}(X),\qquad F_\varepsilon\in\Psi^{2s-2}(X) \end{align*} denote the corresponding regularized operators obtained by applying the same symbol construction to $A_\varepsilon$. The standard pseudodifferential composition and adjoint formulae, applied with the fixed cutoffs from the statement, give the microlocal identity \begin{align*} \frac{1}{i}[P,A_\varepsilon^*A_\varepsilon]=B_\varepsilon^*B_\varepsilon-E_\varepsilon^*E_\varepsilon+R_\varepsilon+F_\varepsilon+\mathcal K_\varepsilon \end{align*} on the chosen conic neighbourhood of $\operatorname{WF}'(A)$, where $\mathcal K_\varepsilon$ is precisely the separated contribution coming from the order $m-1$ skew-adjoint part of $P$. The uniform regularized symbolic decomposition, the operator convergence $B_\varepsilon\to B$, the compatibility estimate controlling $E_\varepsilon$ by $E$ plus the lower Sobolev norm, and the uniform bounds for the symbol seminorms of $A_\varepsilon,B_\varepsilon,E_\varepsilon,R_\varepsilon,F_\varepsilon$ in the relevant compact conic sets are part of the regularizer hypothesis in the statement. They hold in the model regularization because $\Lambda_\varepsilon$ has order $0$ uniformly, converges microlocally to the identity, the symbolic construction is applied with fixed cutoffs, and all displayed conic sets are compact after quotienting by positive fibre rescaling. [/step] [step:Pair the commutator identity with a smooth compactly supported distribution] Assume first that \begin{align*} u\in C_c^\infty(X). \end{align*} All operators are properly supported, so every expression below is a well-defined $L^2$ pairing with respect to the fixed density on $X$. Pairing the regularized commutator identity with $u$ gives \begin{align*} \|B_\varepsilon u\|_{L^2}^2=\|E_\varepsilon u\|_{L^2}^2+\left(\frac{1}{i}[P,A_\varepsilon^*A_\varepsilon]u,u\right)_{L^2}-(R_\varepsilon u,u)_{L^2}-(F_\varepsilon u,u)_{L^2}-K_\varepsilon(u), \end{align*} where \begin{align*} K_\varepsilon(u):=(\mathcal K_\varepsilon u,u)_{L^2}. \end{align*} The sign convention is the one dictated by moving $B_\varepsilon^*B_\varepsilon$ to the left and the remaining terms to the right. By the subprincipal absorption hypothesis, for every $\delta>0$ there is a constant $C_\delta>0$, independent of $\varepsilon$, such that \begin{align*} |K_\varepsilon(u)|\le \delta\|B_\varepsilon u\|_{L^2}^2+C_\delta\left(\|E_\varepsilon u\|_{L^2}^2+\|Gu\|_{L^2}^2+\|u\|_{H^{s-1/2}}^2\right). \end{align*} This is the only place where the possible order $m-1$ skew-adjoint part of $P$ enters. [/step] [step:Rewrite the commutator pairing so that one commutant falls on each factor] Using the definition of the commutator and the $L^2$ adjoint, we have \begin{align*} \left(\frac{1}{i}[P,A_\varepsilon^*A_\varepsilon]u,u\right)_{L^2}=\frac{1}{i}(A_\varepsilon^*A_\varepsilon u,P^*u)_{L^2}-\frac{1}{i}(A_\varepsilon^*A_\varepsilon Pu,u)_{L^2}. \end{align*} Write \begin{align*} P^*u=Pu+(P^*-P)u. \end{align*} The part containing $(P^*-P)u$ is included in the separated skew-adjoint energy contribution already estimated in the previous step. For the two terms containing $Pu$, insert fixed properly supported cutoffs $Q_0,Q_1\in\Psi^0(X)$ such that $Q_0$ is equal to $1$ on $\operatorname{WF}'(A_\varepsilon)$ and $Q_1$ is equal to $1$ on $\operatorname{WF}'(Q_0)$ in the conic region under consideration. The repaired statement includes the required microsupport separation: the microsupports of $1-Q_0$ and $A_\varepsilon$ are disjoint, and the microsupports of $1-Q_1$ and $Q_0$ are disjoint, in the fixed conic region. Hence each term in which one of these cutoffs is replaced by its complement is smoothing microlocally on the displayed region, with a smoothing operator whose seminorms are uniform in $\varepsilon$; after the fixed spatial localization, these errors are uniformly controlled by $\|u\|_{H^{s-1/2}}$. Thus it remains to estimate the Sobolev duality pairing \begin{align*} (A_\varepsilon Q_1Pu,A_\varepsilon Q_0u)_{L^2}. \end{align*} Since $A_\varepsilon$ has order $s-(m-1)/2$, the microlocal hypothesis $Q_1Pu\in H^{s-m+1}(X)$ and Sobolev boundedness of pseudodifferential operators give \begin{align*} \|A_\varepsilon Q_1Pu\|_{H^{-(m-1)/2}}\le C\|Q_1Pu\|_{H^{s-m+1}}, \end{align*} with $C$ independent of $\varepsilon$. [guided] The commutator pairing is the delicate part because it contains $Pu$, and the theorem assumes only microlocal Sobolev regularity of $Pu$, not an a priori $H^s$ bound for $u$. The correct way to use the hypothesis is therefore to split the two factors of $A_\varepsilon^*A_\varepsilon$. We start from the identity \begin{align*} \left(\frac{1}{i}[P,A_\varepsilon^*A_\varepsilon]u,u\right)_{L^2}=\frac{1}{i}(A_\varepsilon^*A_\varepsilon u,P^*u)_{L^2}-\frac{1}{i}(A_\varepsilon^*A_\varepsilon Pu,u)_{L^2}. \end{align*} The adjoint term is rewritten using \begin{align*} P^*u=Pu+(P^*-P)u. \end{align*} The contribution of $(P^*-P)u$ is not automatically lower order: $P^*-P$ may have order $m-1$, and after composition with $A_\varepsilon^*A_\varepsilon$ the resulting energy has the same order as the positive term. This is why the statement includes the subprincipal absorption hypothesis. That hypothesis gives, for every $\delta>0$, \begin{align*} |K_\varepsilon(u)|\le \delta\|B_\varepsilon u\|_{L^2}^2+C_\delta\left(\|E_\varepsilon u\|_{L^2}^2+\|Gu\|_{L^2}^2+\|u\|_{H^{s-1/2}}^2\right). \end{align*} We now focus on the terms containing $Pu$. Choose properly supported cutoffs $Q_0,Q_1\in\Psi^0(X)$ such that $Q_0$ is equal to $1$ on $\operatorname{WF}'(A_\varepsilon)$ and $Q_1$ is equal to $1$ on $\operatorname{WF}'(Q_0)$ in the fixed conic neighbourhood of $\operatorname{WF}'(A)$. These cutoffs convert the phrase “$Pu$ is microlocally in $H^{s-m+1}$ near $\operatorname{WF}'(A)$” into the concrete norm $\|Q_1Pu\|_{H^{s-m+1}}$. The reason the cutoff insertion does not change the main term is the microsupport separation built into the statement. Since $Q_0=1$ on $\operatorname{WF}'(A_\varepsilon)$, the operator $A_\varepsilon(1-Q_0)$ is smoothing on the chosen conic region. Since $Q_1=1$ on $\operatorname{WF}'(Q_0)$, the operator $(1-Q_1)Q_0$ is smoothing there. The same separation holds after taking adjoints because the adjoint has the same operator wavefront relation with source and target interchanged. Therefore every term missed by inserting $Q_0$ and $Q_1$ is a fixed-order smoothing remainder, uniformly in $\varepsilon$, and after spatial localization it is bounded by a fixed multiple of $\|u\|_{H^{s-1/2}}$. With these cutoffs inserted, the main pairing has the form \begin{align*} (A_\varepsilon Q_1Pu,A_\varepsilon Q_0u)_{L^2}. \end{align*} The operator $A_\varepsilon$ has order $s-(m-1)/2$. Therefore, by the standard Sobolev [boundedness theorem](/theorems/181) for pseudodifferential operators, applied uniformly in $\varepsilon$, \begin{align*} \|A_\varepsilon Q_1Pu\|_{H^{-(m-1)/2}}\le C\|Q_1Pu\|_{H^{s-m+1}}. \end{align*} Here the exponent is computed by subtracting the order of $A_\varepsilon$ from the input regularity: \begin{align*} s-m+1-\left(s-\frac{m-1}{2}\right)=-\frac{m-1}{2}. \end{align*} This is exactly the dual Sobolev exponent paired with $H^{(m-1)/2}$. [/guided] [/step] [step:Control the commutant factor by the positive term] The elliptic covering hypothesis in the statement says precisely that the sign construction and the fixed cutoffs cover $\operatorname{WF}'(A)$ by the elliptic regions of $B$, $E$, and $G$ in the amount needed to control $A_\varepsilon$. Applying that hypothesis with $v=u$ gives \begin{align*} \|A_\varepsilon Q_0u\|_{H^{(m-1)/2}}\le C\left(\|B_\varepsilon u\|_{L^2}+\|E_\varepsilon u\|_{L^2}+\|Gu\|_{L^2}+\|u\|_{H^{s-1/2}}\right), \end{align*} where $C$ is independent of $\varepsilon$. The lower-order term is placed at order $s-1/2$ because $A_\varepsilon$ has order $s-(m-1)/2$ and the positive operator has order $s$. Sobolev duality between $H^{-(m-1)/2}$ and $H^{(m-1)/2}$ gives \begin{align*} |(A_\varepsilon Q_1Pu,A_\varepsilon Q_0u)_{L^2}|\le C\|Q_1Pu\|_{H^{s-m+1}}\left(\|B_\varepsilon u\|_{L^2}+\|E_\varepsilon u\|_{L^2}+\|Gu\|_{L^2}+\|u\|_{H^{s-1/2}}\right). \end{align*} Applying [Young's inequality](/theorems/244) with parameter $\delta>0$ to each product yields \begin{align*} |(A_\varepsilon Q_1Pu,A_\varepsilon Q_0u)_{L^2}|\le \delta\|B_\varepsilon u\|_{L^2}^2+C_\delta\left(\|Q_1Pu\|_{H^{s-m+1}}^2+\|E_\varepsilon u\|_{L^2}^2+\|Gu\|_{L^2}^2+\|u\|_{H^{s-1/2}}^2\right). \end{align*} The same estimate applies to the adjoint $Pu$ term, because the cutoffs and the Sobolev orders are the same after taking adjoints. [/step] [step:Bound the elliptic error and the genuinely lower-order remainder] Since $\operatorname{WF}'(R_\varepsilon)$ is contained in the elliptic set of $G$, choose a properly supported parametrix \begin{align*} S\in\Psi^{-s}(X) \end{align*} for $G$ on a conic neighbourhood of $\operatorname{WF}'(R_\varepsilon)$, uniformly in $\varepsilon$, and a smoothing remainder $T_\varepsilon$ microlocally near $\operatorname{WF}'(R_\varepsilon)$ such that \begin{align*} R_\varepsilon=R_\varepsilon S G+T_\varepsilon \end{align*} microlocally on the fixed conic region. Because \begin{align*} R_\varepsilon S\in\Psi^{s-1}(X), \end{align*} choose fixed properly supported cutoffs $Q_R,Q_R'\in\Psi^0(X)$ with $Q_R=1$ on $\operatorname{WF}'(R_\varepsilon)$ and $Q_R'=1$ on $\operatorname{WF}'(Q_R)$ in the same conic region. The smoothing errors from inserting these cutoffs are included in the lower Sobolev term. Then \begin{align*} (R_\varepsilon SGu,u)_{L^2}=(Q_RGu,(R_\varepsilon S)^*Q_R'u)_{L^2}+O(\|u\|_{H^{s-1/2}}^2). \end{align*} The adjoint $(R_\varepsilon S)^*$ has order $s-1$, so Sobolev boundedness gives \begin{align*} \|(R_\varepsilon S)^*Q_R'u\|_{L^2}\le C\|Q_R'u\|_{H^{s-1}}. \end{align*} Since $H^{s-1/2}(X)$ continuously embeds into $H^{s-1}(X)$, Cauchy's inequality gives \begin{align*} |(R_\varepsilon u,u)_{L^2}|\le C\left(\|Gu\|_{L^2}^2+\|u\|_{H^{s-1/2}}^2\right). \end{align*} The constants are uniform in $\varepsilon$ because the symbol seminorms of $R_\varepsilon$ are uniformly bounded, and the smoothing remainder is included in the lower Sobolev term. Next, since \begin{align*} F_\varepsilon\in\Psi^{2s-2}(X), \end{align*} the operator $F_\varepsilon$ maps $H^{s-1/2}(X)$ continuously into $H^{-s+3/2}(X)$ and, after insertion of the fixed compact microlocal cutoffs, its quadratic form is bounded by \begin{align*} |(F_\varepsilon u,u)_{L^2}|\le C\|u\|_{H^{s-1/2}}^2. \end{align*} The constant $C$ is independent of $\varepsilon$ because the symbol seminorms of $F_\varepsilon$ are uniformly bounded. Thus both error terms are controlled by the right-hand side allowed in the theorem. [/step] [step:Absorb the small positive-term losses and obtain the uniform estimate] Combining the preceding bounds gives, for every sufficiently small $\delta>0$, \begin{align*} \|B_\varepsilon u\|_{L^2}^2\le C_0\|E_\varepsilon u\|_{L^2}^2+C_0\|Gu\|_{L^2}^2+C_0\|Q_1Pu\|_{H^{s-m+1}}^2+C_0\|u\|_{H^{s-1/2}}^2+C_0\delta\|B_\varepsilon u\|_{L^2}^2. \end{align*} Choose $\delta>0$ so that \begin{align*} C_0\delta\le \frac{1}{2}. \end{align*} Moving the final term on the right to the left gives \begin{align*} \|B_\varepsilon u\|_{L^2}^2\le C\left(\|E_\varepsilon u\|_{L^2}^2+\|Gu\|_{L^2}^2+\|Q_1Pu\|_{H^{s-m+1}}^2+\|u\|_{H^{s-1/2}}^2\right), \end{align*} where $C>0$ is independent of $\varepsilon$. By the regularizer compatibility hypothesis in the repaired statement, \begin{align*} \|E_\varepsilon u\|_{L^2}\le C\left(\|Eu\|_{L^2}+\|u\|_{H^{s-1/2}}\right), \end{align*} with $C$ independent of $\varepsilon$. Substituting this estimate and identifying $\|Q_1Pu\|_{H^{s-m+1}}$ with the fixed-cutoff representative of $\|Pu\|_{H^{s-m+1}(\operatorname{WF}'(A))}$ yields \begin{align*} \|B_\varepsilon u\|_{L^2}^2\le C\left(\|Eu\|_{L^2}^2+\|Gu\|_{L^2}^2+\|Pu\|_{H^{s-m+1}(\operatorname{WF}'(A))}^2+\|u\|_{H^{s-1/2}}^2\right). \end{align*} [/step] [step:Pass from the regularized estimate to the unregularized positive term] The estimate from the previous step is uniform in $\varepsilon$. Hence the family \begin{align*} (B_\varepsilon u)_{0<\varepsilon\le 1} \end{align*} is bounded in $L^2(X)$ for smooth compactly supported $u$. By weak compactness of bounded sets in the [Hilbert space](/page/Hilbert%20Space) $L^2(X)$, there exist a sequence $\varepsilon_j\to 0$ and an element \begin{align*} v\in L^2(X) \end{align*} such that \begin{align*} B_{\varepsilon_j}u\rightharpoonup v \end{align*} weakly in $L^2(X)$. For every [test function](/page/Test%20Function) \begin{align*} \phi\in C_c^\infty(X), \end{align*} the operator convergence $B_\varepsilon\to B$ on localized distributions, which is part of the regularizer hypothesis in the repaired statement, and the uniform proper support imply \begin{align*} (B_{\varepsilon_j}u,\phi)_{L^2}\to (Bu,\phi)_{\mathcal D',C_c^\infty}. \end{align*} Therefore $v=Bu$ as a distribution. Since $v\in L^2(X)$, we obtain \begin{align*} Bu\in L^2(X). \end{align*} Lower semicontinuity of the $L^2$ norm under [weak convergence](/page/Weak%20Convergence) gives \begin{align*} \|Bu\|_{L^2}^2\le \liminf_{j\to\infty}\|B_{\varepsilon_j}u\|_{L^2}^2. \end{align*} Using the uniform estimate for $B_{\varepsilon_j}u$ proves \begin{align*} \|Bu\|_{L^2}^2\le C\left(\|Eu\|_{L^2}^2+\|Gu\|_{L^2}^2+\|Pu\|_{H^{s-m+1}(\operatorname{WF}'(A))}^2+\|u\|_{H^{s-1/2}}^2\right) \end{align*} for smooth compactly supported $u$. [/step] [step:Extend the estimate to localized distributions] Let now \begin{align*} u\in\mathcal D'(X) \end{align*} satisfy the hypotheses of the theorem after applying the fixed properly supported localization cutoff. Choose the smoothing approximation from the approximation hypothesis in the statement: \begin{align*} u_j\in C_c^\infty(X). \end{align*} Thus $u_j\to u$ in $\mathcal D'(X)$ after the same spatial localization, and the fixed localized quantities satisfy \begin{align*} Eu_j\to Eu \quad \text{in } L^2_{\mathrm{loc}}(X), \end{align*} \begin{align*} Gu_j\to Gu \quad \text{in } L^2_{\mathrm{loc}}(X), \end{align*} \begin{align*} Q_1Pu_j\to Q_1Pu \quad \text{in } H^{s-m+1}(X), \end{align*} and \begin{align*} u_j\to u \quad \text{in the localized } H^{s-1/2} \text{ norm appearing in the estimate}. \end{align*} Apply the smooth estimate to $u_j$. The displayed convergences imply that the right-hand side for $u_j$ converges to the right-hand side for $u$. Thus $(Bu_j)$ is bounded in $L^2_{\mathrm{loc}}(X)$. Passing to a weakly convergent subsequence on each compact set and testing against $C_c^\infty$ functions identifies the weak limit with $Bu$ as a distribution. Hence \begin{align*} Bu\in L^2_{\mathrm{loc}}(X). \end{align*} The same weak lower semicontinuity argument gives the stated fixed-cutoff estimate for $u$. This completes the proof. [/step]