## Hardy inequalities

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- by Andreas Wannebo PDF
- Proc. Amer. Math. Soc.
**109**(1990), 85-95 Request permission

## Abstract:

Here the following Hardy inequalities are studied \[ \sum \limits _{k = 0}^{m - 1} {\int {\frac {{|{\nabla ^k}u{|^p}{d_{\partial \Omega }}{{(x)}^t}}}{{{d_{\partial \Omega }}{{(x)}^{(m - k)p}}}}dx \leq {A_\Omega }\int {|{\nabla ^m}u{|^p}{d_{\partial \Omega }}{{(x)}^t}dx} } } \] for $u \in C_0^\infty (\Omega )$, an open proper subset of ${{\mathbf {R}}^N}$ and $t < {t_0}$, some small positive ${t_0}$. This inequality has previously been shown to hold for bounded Lipschitz domains. The question discussed is, How general can $\Omega$ be and allow the same inequality? A sufficient condition is given in the form of a local Maz’ja capacity condition. In ${{\mathbf {R}}^2}$, or generally if $p > N - 1$, this is satisfied for any $\Omega$ which is deformable to a point. Furthermore, if $p > N$ the condition is satisfied for all $\Omega$.## References

- Alano Ancona,
*Une propriété des espaces de Sobolev*, C. R. Acad. Sci. Paris Sér. I Math.**292**(1981), no. 9, 477–480 (French, with English summary). MR**612540** - Haïm Brézis and Felix E. Browder,
*Some properties of higher order Sobolev spaces*, J. Math. Pures Appl. (9)**61**(1982), no. 3, 245–259 (1983). MR**690395** - G. H. Hardy, J. E. Littlewood, and G. Pólya,
*Inequalities*, Cambridge, at the University Press, 1952. 2d ed. MR**0046395** - Lars Inge Hedberg,
*Two approximation problems in function spaces*, Ark. Mat.**16**(1978), no. 1, 51–81. MR**499137**, DOI 10.1007/BF02385982 - Alois Kufner,
*Weighted Sobolev spaces*, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. Translated from the Czech. MR**802206** - V. G. Maz′ja,
*The $(p,\,l)$-capacity, imbedding theorems and the spectrum of a selfadjoint elliptic operator*, Izv. Akad. Nauk SSSR Ser. Mat.**37**(1973), 356–385 (Russian). MR**0338766** - Vladimir G. Maz’ja,
*Sobolev spaces*, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR**817985**, DOI 10.1007/978-3-662-09922-3 - V. G. Maz′ja and V. P. Havin,
*A nonlinear potential theory*, Uspehi Mat. Nauk**27**(1972), no. 6, 67–138. MR**0409858** - Norman G. Meyers,
*Continuity of Bessel potentials*, Israel J. Math.**11**(1972), 271–283. MR**301216**, DOI 10.1007/BF02789319 - Jindřich Nečas,
*Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)**16**(1962), 305–326 (French). MR**163054** - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095** - Edward W. Stredulinsky,
*Weighted inequalities and degenerate elliptic partial differential equations*, Lecture Notes in Mathematics, vol. 1074, Springer-Verlag, Berlin, 1984. MR**757718**, DOI 10.1007/BFb0101268
A. Wannebo,

*Poincaré type inequalities for a cube and Hardy inequalities for a domain*(in preparation).

## Additional Information

- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**109**(1990), 85-95 - MSC: Primary 26D10; Secondary 46E35
- DOI: https://doi.org/10.1090/S0002-9939-1990-1010807-1
- MathSciNet review: 1010807