On the behavior of weak convergence under nonlinearities and applications
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- by Diego R. Moreira and Eduardo V. Teixeira PDF
- Proc. Amer. Math. Soc. 133 (2005), 1647-1656 Request permission
Abstract:
This paper provides a sufficient condition to guarantee the stability of weak limits under nonlinear operators acting on vector-valued Lebesgue spaces. This nonlinear framework places the weak convergence in perspective. Such an approach allows short and insightful proofs of important results in Functional Analysis such as: weak convergence in $L^\infty$ implies strong convergence in $L^p$ for all $1\le p < \infty$, weak convergence in $L^1$ vs. strong convergence in $L^1$ and the Brezis-Lieb theorem. The final goal is to use this framework as a strategy to grapple with a nonlinear weak spectral problem on $W^{1,p}$.References
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Additional Information
- Diego R. Moreira
- Affiliation: Department of Mathematics, University of Texas at Austin, RLM 12.128, Austin, Texas 78712-1082
- Email: dmoreira@math.utexas.edu
- Eduardo V. Teixeira
- Affiliation: Department of Mathematics, University of Texas at Austin, RLM 9.136, Austin, Texas 78712-1082
- MR Author ID: 710372
- Email: teixeira@math.utexas.edu
- Received by editor(s): April 24, 2003
- Published electronically: December 21, 2004
- Additional Notes: The second author is grateful for the financial support by CNPq - Brazil
- Communicated by: Jonathan M. Borwein
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 1647-1656
- MSC (2000): Primary 46B03, 46B10, 46B20
- DOI: https://doi.org/10.1090/S0002-9939-04-07876-1
- MathSciNet review: 2120260