A compensated compactness theorem for pseudodifferential operators on vector bundles

TL;DR

This paper extends the classical compensated compactness theory to the setting of pseudo-differential operators on vector bundles over semi-Riemannian manifolds, providing a microlocal and geometric framework for convergence results.

Contribution

It introduces a new compensated compactness theorem for pseudo-differential operators on vector bundles, generalizing Murat--Tartar theory to variable-coefficient, higher-order operators on manifolds.

Findings

Established a microlocal compensated compactness theorem for pseudo-differential operators.

Extended classical theory from Euclidean spaces to semi-Riemannian manifolds.

Proved convergence of quadratic forms under geometric and microlocal conditions.

Abstract

We establish a compensated compactness theorem in the microlocal and geometric analytic framework. For a weakly $L^2_{\rm loc}$-convergent sequence of sections of a vector bundle over a semi-Riemannian manifold whose image under a pseudo-differential operator $\mathscr{A}$ of order $s>0$ is precompact in $H^{-s}_{\rm loc}$, 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 $\mathscr{A}$. This extends the classical Murat--Tartar theory of compensated compactness from constant-coefficient first-order differential constraints on Euclidean spaces to variable-coefficient pseudo-differential constraints of arbitrary order on semi-Riemannian manifolds.