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 ${\mathcal{D}}_L^{\prime r}(M;E)$ of distributional sections of the smooth complex vector bundle $E\to M$ whose Sobolev wave front set of order $r\in \mathbb{R}$ lies in the closed conic subset $L$ of $T^{\ast }M\backslash 0$. We introduce a locally convex topology on it to study the continuity of the pullback by smooth maps and generalise the result of Hörmander about the pullback on the space of distributions with ${\mathcal{C}}^{\infty }$ wave front set in $L$. We employ an idea of Gérard [18] to extend the Kolmogorov-Riesz compactness theorem to ${\mathcal{D}}_L^{\prime r}(M;E)$ and we characterise its relatively compact subsets. We study the continuity properties of pseudo-differential operators when acting on ${\mathcal{D}}_L^{\prime r}(M;E)$, $r\in \mathbb{R}$, and we generalise the Rellich's lemma. As an application of our results, we extend the microlocal defect measures of Gérard and Tartar to sequences in ${\mathcal{D}}_L^{\prime 0}(M;E)$ and we show a microlocal variant of the compensated compactness theorem.

85 pages