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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hardy inequalities with optimal constants and remainder terms


Authors: Filippo Gazzola, Hans-Christoph Grunau and Enzo Mitidieri
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2149-2168
MSC (2000): Primary 46E35; Secondary 35B50, 35J40
Published electronically: December 9, 2003
MathSciNet review: 2048513
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces $W_0^{1,p}$ and in higher-order Sobolev spaces on a bounded domain $\Omega\subset\mathbb{R} ^n$ can be refined by adding remainder terms which involve $L^p$ norms. In the higher-order case further $L^p$ norms with lower-order singular weights arise. The case $1<p<2$ being more involved requires a different technique and is developed only in the space $W_0^{1,p}$.


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Additional Information

Filippo Gazzola
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy
Email: gazzola@mate.polimi.it

Hans-Christoph Grunau
Affiliation: Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120, D-39016 Magdeburg, Germany
Email: Hans-Christoph.Grunau@mathematik.uni-magdeburg.de

Enzo Mitidieri
Affiliation: Dipartimento di Scienze Matematiche, Via A. Valerio 12/1, Università degli Studi di Trieste, I-34100 Trieste, Italy
Email: mitidier@univ.trieste.it

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03395-6
PII: S 0002-9947(03)03395-6
Received by editor(s): June 20, 2000
Received by editor(s) in revised form: May 8, 2003
Published electronically: December 9, 2003
Article copyright: © Copyright 2003 American Mathematical Society