Tohoku Mathematical Journal

Hardy type inequalities on balls

Shuji Machihara, Tohru Ozawa, and Hidemitsu Wadade

Full-text: Open access

Abstract

Hardy type inequalities are presented on balls with radius $R$ at the origin in ${\boldsymbol R}^n$ with $n=2$ at least. A special attention is paid on the behavior of functions on the boundary.

Article information

Source
Tohoku Math. J. (2), Volume 65, Number 3 (2013), 321-330.

Dates
First available in Project Euclid: 12 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1378991018

Digital Object Identifier
doi:10.2748/tmj/1378991018

Mathematical Reviews number (MathSciNet)
MR3102537

Zentralblatt MATH identifier
1281.26015

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Keywords
Hardy inequalities

Citation

Machihara, Shuji; Ozawa, Tohru; Wadade, Hidemitsu. Hardy type inequalities on balls. Tohoku Math. J. (2) 65 (2013), no. 3, 321--330. doi:10.2748/tmj/1378991018. https://projecteuclid.org/euclid.tmj/1378991018


Export citation

References

  • Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc. 130 (2002), 489–505.
  • W. Beckner, Pitt's inequality with sharp convolution estimates, Proc. Amer. Math. Soc. 136 (2008), 1871–1885.
  • J. Bergh and J. Löfström, Interpolation spaces, An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.
  • H. Brezis and M. Marcus, Hardy's inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 217–237.
  • D. E. Edmunds and H. Triebel, Sharp Sobolev embeddings and related Hardy inequalities: the critical case, Math. Nachr. 207 (1999), 79–92.
  • G. B. Folland, Real analysis, second edition, Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1999.
  • J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), 441–476.
  • I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$, Comm. Math. Phys. 53 (1977), 285–294.
  • H. Kalf and J. Walter, Strongly singular potentials and essential self-adjointness of singular elliptic operators in $C_0^{\infty}({\bf R}^n\backslash \{0\})$, J. Functional Analysis 10 (1972), 114–130.
  • O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.
  • J. Leray, Etude de diverses équations integrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl. 12 (1933), 1–82.
  • J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972.
  • A. Matsumura and N. Yamagata, Global weak solutions of the Navier-Stokes equations for multidimensional compressible flow subject to large external potential forces, Osaka J. Math. 38 (2001), 399–418.
  • T. Ozawa and H. Sasaki, Inequalities associated with dilations, Commun. Contemp. Math. 11 (2009), 265–277.
  • L. Pick, Optimal Sobolev embeddings, Nonlinear analysis, function spaces and applications, Vol. 6 (Prague, 1998), 156–199, Acad. Sci. Czech Repub., Prague, 1999.
  • M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
  • H. Triebel, Sharp Sobolev embeddings and related Hardy inequalities: the sub-critical case, Math. Nachr. 208 (1999), 167–178.
  • J. Zhang, Extensions of Hardy inequality, J. Inequal. Appl. (2006), Art. ID 69379.