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

 

A unified approach to improved $L^p$ Hardy inequalities with best constants


Authors: G. Barbatis, S. Filippas and A. Tertikas
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2169-2196
MSC (2000): Primary 35J20, 26D10; Secondary 46E35, 35Pxx
Published electronically: December 9, 2003
MathSciNet review: 2048514
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Abstract: We present a unified approach to improved $L^p$ Hardy inequalities in $\mathbf{R}^N$. We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension $1<k<N$. In our main result, we add to the right hand side of the classical Hardy inequality a weighted $L^p$ norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted $L^q$ norms, $q \neq p$.


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Additional Information

G. Barbatis
Affiliation: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Email: gbarbati@cc.uoi.gr

S. Filippas
Affiliation: Department of Applied Mathematics, University of Crete, 71409 Heraklion, Greece
Email: filippas@tem.uoc.gr

A. Tertikas
Affiliation: Department of Mathematics, University of Crete, 71409 Heraklion, Greece and Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Greece
Email: tertikas@math.uoc.gr

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03389-0
PII: S 0002-9947(03)03389-0
Keywords: Hardy inequalities, best constants, distance function, weighted norms
Received by editor(s): February 28, 2001
Published electronically: December 9, 2003
Article copyright: © Copyright 2003 American Mathematical Society