Spaces of distributions with Sobolev wave front in a fixed conic set: compactness, pullback by smooth maps and the compensated compactness theorem
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| Format: | Preprint |
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2024
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| _version_ | 1866913473675395072 |
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| author | Pilipović, Stevan Prangoski, Bojan |
| author_facet | Pilipović, Stevan Prangoski, Bojan |
| contents | 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 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 $\mathcal{D}'^r_L(M;E)$ 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 Gérard and Tartar to sequences in $\mathcal{D}'^0_L(M;E)$ and we show a microlocal variant of the compensated compactness theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_10741 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spaces of distributions with Sobolev wave front in a fixed conic set: compactness, pullback by smooth maps and the compensated compactness theorem Pilipović, Stevan Prangoski, Bojan Analysis of PDEs Differential Geometry Functional Analysis 46E35, 58J40 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 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 $\mathcal{D}'^r_L(M;E)$ 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 Gérard and Tartar to sequences in $\mathcal{D}'^0_L(M;E)$ and we show a microlocal variant of the compensated compactness theorem. |
| title | Spaces of distributions with Sobolev wave front in a fixed conic set: compactness, pullback by smooth maps and the compensated compactness theorem |
| topic | Analysis of PDEs Differential Geometry Functional Analysis 46E35, 58J40 |
| url | https://arxiv.org/abs/2408.10741 |