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

Abstract: We consider the space $\mathcal{D}'^r_L(M;E)$ of distributional sections of the smooth complex vector bundle $E\rightarrow M$ whose Sobolev wave front set of order $r\in\mathbb{R}$ lies in the closed conic subset $L$ of $T^*M\backslash0$. 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 $\mathcal{C}^{\infty}$ wave front set in $L$. We employ an idea of G\'erard [18] to extend the Kolmogorov-Riesz compactness theorem to $\mathcal{D}'^r_L(M;E)$ and we characterise its relatively compact subsets. We study the continuity properties of pseudo-differential operators when acting on $\mathcal{D}'^r_L(M;E)$, $r\in\mathbb{R}$, 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 $\mathcal{D}'^0_L(M;E)$ and we show a microlocal variant of the compensated compactness theorem.