A compensated compactness theorem for pseudodifferential operators on vector bundles

We establish a compensated compactness theorem in the microlocal and geometric analytic framework. For a weakly $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$-convergent sequence of sections of a vector bundle over a semi-Riemannian manifold whose image under a pseudo-differential operator $\mathcal{A}$ of order $s>0$ is precompact in $H_{\mathrm{l}\mathrm{o}\mathrm{c}}^{-s}$, we show that a quadratic form $Q$ acting on this sequence converges in the distributional sense, provided that $Q$ vanishes on the operator cone of $\mathcal{A}$. This extends the classical Murat–Tartar theory of compensated compactness from constant-coefficient first-order differential constraints on Euclidean spaces to variable-coefficient pseudo-differential constraints of arbitrary order on semi-Riemannian manifolds.