Reflexivity of the space of transversal distributions

Kališnik, Jure (author)

For any smooth, Hausdorff and second-countable manifold $N$ one can define the Fréchet space ${\mathcal C}^{\infty}(N)$ of smooth functions on $N$ and its strong dual ${\cal E}'(N)$ of compactly supported distributions on $N$. It is a standard result that the strong dual of ${\cal E}'(N)$ is naturally isomorphic to ${\mathcal C}^{\infty}(N)$, which implies that both ${\mathcal C}^{\infty}(N)$ and ${\cal E}'(N)$ are reflexive locally convex spaces. In this paper we generalise that result to the setting of transversal distributions on the total space of a surjective submersion $\pi : P\to M$. We show that the strong ${\mathcal C}^{\infty}_c(M)$-dual of the space ${\cal E}'_{\pi} (P)$ of $\pi$-transversal distributions is naturally isomorphic to the ${\mathcal C}^{\infty}_c(M)$-module ${\mathcal C}^{\infty}(P)$.

• Language: English
• Keywords: distributions with compact support, Fréchet spaces, transversal distributions, homomorphisms of modules, reflexive modules
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2023
• Number of pages: 20 str.
• Numbering: Vol. 33, iss. 10, art. 331
• PID: 20.500.12556/RUL-153557
• UDC: 517.9
• ISSN on article: 1050-6926
• DOI: 10.1007/s12220-023-01390-y
• COBISS.SI-ID: 178959107
• Publication date in RUL: 15.01.2024

Record is a part of a journal

• Title: The journal of geometric analysis
• Shortened title: J. geom. anal.
• Publisher: Springer Nature, Mathematica Josephina, Inc.
• ISSN: 1050-6926
• COBISS.SI-ID: 30685696

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.