Spaces of distributions with Sobolev wave front in a fixed conic set: compactness, pullback by smooth maps and the compensated compactness theorem

We consider the space D'ʳL(M;E) of distributional sections of the smooth complex vector bundle E→ M whose Sobolev wave front set of order r lies in the closed conic subset L of T^*M0. We introduce a locally convex topology on it to study the continuity of the pullback by smooth maps and generalise the result of H\"ormander about the pullback on the space of distributions with C∞ wave front set in L. We employ an idea of G\'erard [18] to extend the Kolmogorov-Riesz compactness theorem to D'ʳL(M;E) and we characterise its relatively compact subsets. We study the continuity properties of pseudo-differential operators when acting on D'ʳL(M;E), r, and we generalise the Rellich's lemma. As an application of our results, we extend the microlocal defect measures of G\'erard and Tartar to sequences in D'⁰L(M;E) and we show a microlocal variant of the compensated compactness theorem.