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Inequalities pp 523-527 | Cite as

A Relation Between Pointwise Convergence of Functions and Convergence of Functionals

  • Haïm Brezis
  • Elliott Lieb

Abstract

We show that if f n is a sequence of uniformly L p-bounded functions on a measure space, and if f nf pointwise a.e., then lim for all 0 < p < ∞. This result is also generalized in Theorem 2 to some functional other than the L p norm, namely → 0 for suitable j: C → C and a suitable sequence f n. A brief discussion is given of the usefulness of this result in variational problems.

Key words

Convergence of functional pointwise convergence Lp spaces 

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References

  1. 1.
    E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math, (to appear).Google Scholar
  2. 2.
    H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math, (to appear).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Haïm Brezis
    • 1
  • Elliott Lieb
    • 1
    • 2
  1. 1.Département de MathématiquesUniversité Paris VIParis Cedex 05France
  2. 2.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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