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A geometrical version of Hardy's inequality for $\stackrel{\circ}{\textrm{W}}{}^{1,p}(\Omega)$

Author: Jesper Tidblom
Journal: Proc. Amer. Math. Soc. 132 (2004), 2265-2271
MSC (2000): Primary 35P99; Secondary 35P20, 47A75, 47B25
Published electronically: March 25, 2004
MathSciNet review: 2052402
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Abstract: The aim of this article is to prove a Hardy-type inequality, concerning functions in $\stackrel{\circ}{\textrm{W}}{}^{1,p}(\Omega)$ for some domain $\Omega \subset R^n$, involving the volume of $\Omega$ and the distance to the boundary of $\Omega$. The inequality is a generalization of a recently proved inequality by M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and A. Laptev (2002), which dealt with the special case $p=2$.

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  • 1. G. Barbatis, S. Filippas, and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, preprint, December 2000.
  • 2. H. Brezis and M. Marcus, Hardy's inequalities revisited, dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) vol. 25 (1997) no. 1-2 (1998), pp. 217-237. MR 99m:46075
  • 3. E. B. Davies, A review of Hardy inequalities, Oper. Theory Adv. Appl., vol. 110 (1998), pp. 55-67. MR 2001f:35166
  • 4. E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1989. MR 92a:35035
  • 5. E. B. Davies, The Hardy constant, Quart. J. Math. Oxford (2), vol. 46 (1995), pp. 417-431. MR 97b:46041
  • 6. E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 96h:47056
  • 7. G. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1952. MR 13:727e
  • 8. G. Hardy, Note on a theorem of Hilbert, Math. Zeitschrift (6) (1920), pp. 314-317.
  • 9. M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and A. Laptev, A geometrical version of Hardy's inequality, J. Funct. Anal. 189 (2002), no. 2, 539-548. MR 2003c:26022
  • 10. E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 1997. MR 98b:00004
  • 11. T. Matskewich and P. E. Sobolevskii, The best possible constant in a generalized Hardy's inequality for convex domains in $\mathbb{R} ^n$, Nonlinear Anal., vol. 28 (1997), pp. 1601-1610. MR 98a:26019
  • 12. V. G. Maz'ja, Sobolev Spaces, Springer-Verlag, Berlin, 1985. MR 87g:46056
  • 13. B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Group UK Limited, London, 1990. MR 92b:26028

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Additional Information

Jesper Tidblom
Affiliation: Department of Mathematics, University of Stockholm, 106 91 Stockholm, Sweden

Received by editor(s): January 28, 2002
Published electronically: March 25, 2004
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2004 American Mathematical Society

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