Abstract
We show that if f n is a sequence of uniformly L p-bounded functions on a measure space, and if f n → f 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.
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References
E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math, (to appear).
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math, (to appear).
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© 2002 Springer-Verlag Berlin Heidelberg
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Brezis, H., Lieb, E. (2002). A Relation Between Pointwise Convergence of Functions and Convergence of Functionals. In: Loss, M., Ruskai, M.B. (eds) Inequalities. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55925-9_42
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DOI: https://doi.org/10.1007/978-3-642-55925-9_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62758-3
Online ISBN: 978-3-642-55925-9
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