Topology of the space of conormal distributions
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2023
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| author | López, Jesús A. Álvarez Kordyukov, Yuri A. Leichtnam, Eric |
| author_facet | López, Jesús A. Álvarez Kordyukov, Yuri A. Leichtnam, Eric |
| contents | Given a closed manifold $M$ and a closed regular submanifold $L$, consider the corresponding locally convex space $I=I(M,L)$ of conormal distributions, with its natural topology, and the strong dual $I'=I'(M,L)=I(M,L;Ω)'$ of the space of conormal densities. It is shown that $I$ is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and $I'$ is a complete Montel space, which is a projective limit of bornological barreled spaces. In the case of codimension one, similar properties and additional descriptions are proved for the subspace $K\subset I$ of conormal distributions supported in $L$ and for its strong dual $K'$. We construct a locally convex Hausdoff space $J$ and a continuous linear map $I\to J$ such that the sequence $0\to K\to I\to J\to 0$ as well as the transpose sequence $0\to J'\to I'\to K'\to 0$ are short exact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that $I\cap I'=C^\infty(M)$ in the space of distributions. In another publication, these results are applied to prove a Lefschetz trace formula for a simple foliated flow $ϕ=\{ϕ^t\}$ on a compact foliated manifold $(M,F)$. It describes a Lefschetz distribution $L_{\text{\rm dis}}(ϕ)$ defined by the induced action $ϕ^*=\{ϕ^{t\,*}\}$ on the reduced cohomologies $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by $ϕ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_00798 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Topology of the space of conormal distributions López, Jesús A. Álvarez Kordyukov, Yuri A. Leichtnam, Eric Functional Analysis 46F05, 46A13, 46M40 Given a closed manifold $M$ and a closed regular submanifold $L$, consider the corresponding locally convex space $I=I(M,L)$ of conormal distributions, with its natural topology, and the strong dual $I'=I'(M,L)=I(M,L;Ω)'$ of the space of conormal densities. It is shown that $I$ is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and $I'$ is a complete Montel space, which is a projective limit of bornological barreled spaces. In the case of codimension one, similar properties and additional descriptions are proved for the subspace $K\subset I$ of conormal distributions supported in $L$ and for its strong dual $K'$. We construct a locally convex Hausdoff space $J$ and a continuous linear map $I\to J$ such that the sequence $0\to K\to I\to J\to 0$ as well as the transpose sequence $0\to J'\to I'\to K'\to 0$ are short exact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that $I\cap I'=C^\infty(M)$ in the space of distributions. In another publication, these results are applied to prove a Lefschetz trace formula for a simple foliated flow $ϕ=\{ϕ^t\}$ on a compact foliated manifold $(M,F)$. It describes a Lefschetz distribution $L_{\text{\rm dis}}(ϕ)$ defined by the induced action $ϕ^*=\{ϕ^{t\,*}\}$ on the reduced cohomologies $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by $ϕ$. |
| title | Topology of the space of conormal distributions |
| topic | Functional Analysis 46F05, 46A13, 46M40 |
| url | https://arxiv.org/abs/2304.00798 |