Linear continuous division for exterior and interior products
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- by P. Domański and B. Jakubczyk PDF
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Abstract:
We consider the complex \[ \begin {CD} 0@>>> {\Lambda _ 0(M;E)} @>{\partial _ \omega }>> {\Lambda _ 1(M;E)} @>{\partial _\omega }>>{\dots } @>{\partial _\omega }>> {\Lambda _ m(M;E)}, \end {CD} \] where $E$ is a finite-dimensional vector bundle over a suitable differential manifold $M$, $\Lambda _ q(M;E)$ denotes the space of all smooth or real analytic or holomorphic sections of the $q$-exterior product of $E$ and $\partial _\omega (\eta ):=\omega \wedge \eta$ for $\omega \in \Lambda _ 1(M;E)$. We give sufficient and necessary conditions for the above complex to be exact and, in smooth and holomorphic cases, we give sufficient conditions for its splitting, i.e., for existence of linear continuous right inverse operators for $\partial _\omega :\Lambda _ q(M;E)\to \operatorname {Im} \partial _\omega \subseteq \Lambda _{q+1}(M;E)$. Analogous results are obtained whenever $M$ is replaced by a suitable closed subset $X$ or $\partial _\omega$ are replaced by the interior product operators $\partial _ Z$, $\partial _ Z(\eta ):=Z\rfloor \eta$ for a given section $Z$ of the dual bundle $E^ *$.References
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Additional Information
- P. Domański
- Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University and Institute of Mathematics (Poznań branch), Polish Academy of Sciences, Umultowska 87, 61-614 Poznań, Poland
- Email: domanski@math.amu.edu.pl
- B. Jakubczyk
- Affiliation: Institute of Applied Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland On leave from: Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland
- Email: B.Jakubczyk@impan.gov.pl
- Received by editor(s): May 7, 2002
- Published electronically: May 9, 2003
- Additional Notes: The research of the second named author was partially supported by the Committee for Scientific Research, Poland, grant KBN 2P03A 03516
- Communicated by: Jozef Dodziuk
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3163-3175
- MSC (2000): Primary 46E10, 58A10
- DOI: https://doi.org/10.1090/S0002-9939-03-07107-7
- MathSciNet review: 1992857