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 l o c 2 L^2_{loc} -convergent sequence of sections of a vector bundle over a semi-Riemannian manifold whose image under a pseudodifferential operator A \mathscr {A} of order s > 0 s>0 is precompact in H l o c − s H^{-s}_{loc} , we show that a quadratic form Q Q acting on this sequence converges in the distributional sense, provided that Q Q vanishes on the operator cone of A \mathscr {A} . This extends the classical Murat–Tartar theory of compensated compactness from constant-coefficient first-order differential constraints on Euclidean spaces to variable-coefficient pseudodifferential constraints of arbitrary order on semi-Riemannian manifolds.