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

TL;DR

This paper develops a framework for analyzing distributions with Sobolev wave front sets in fixed conic regions, extending classical theorems like compactness, pullback, and Rellich's lemma to this microlocal setting.

Contribution

It introduces a new locally convex topology on these distribution spaces, generalizes key theorems, and applies them to microlocal defect measures and compensated compactness.

Findings

Extended H"ormander's pullback theorem to Sobolev wave front sets.

Generalized Kolmogorov-Riesz compactness theorem for these distributions.

Proved a microlocal version of 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.