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On the behavior of weak convergence under nonlinearities and applications


Authors: Diego R. Moreira and Eduardo V. Teixeira
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
Published electronically: December 21, 2004
MathSciNet review: 2120260
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. R. Adams, Sobolev spaces, Academic Press, New York (1975). MR 0450957 (56:9247)
  • 2. H. Brezis, Analyse fonctionnelle, Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, 1983. MR 0697382 (85a:46001)
  • 3. H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490. MR 0699419 (84e:28003)
  • 4. J. Diestel and J. J. Uhl, Jr. , Vector measures, Mathematical surveys, No 15. AMS 1977. MR 0453964 (56:12216)
  • 5. N. Dunford and J. Schwartz, Linear operator, Interscience Publishers, Inc., New York, second printing (1964). MR 0117523 (22:8302)
  • 6. R. Lucchetti and F. Patrone, On Nemytskii's operator and its application to the lower semicontinuity of integral functionals, Indiana University Mathematics Journal, Vol. 29 No 5 (1980). MR 0589437 (82i:47104)
  • 7. H. P. Rosenthal, A characterization of Banach spaces containing $l^1$, Proc. Nat. Acad. Sci. U.S.A., 71 (1974). MR 0358307 (50:10773)
  • 8. M. Talagrand, Weak Cauchy sequence in $L^1(E)$, American Journal of Mathematics, 106 No 3 (1984). MR 0745148 (85j:46062)
  • 9. T. Zolezzi, On Weak Convergence in $L^\infty$, Indiana Univesity Mathematics Journal, Vol. 23, No 8 (1974). MR 0328576 (48''6918)

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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
Email: teixeira@math.utexas.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07876-1
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
Article copyright: © Copyright 2004 American Mathematical Society

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