Skip to main content
SpringerLink
Log in

A Note on the Scale Invariant Structure of Critical Hardy Inequalities

  • Conference paper
  • First Online:
Geometric Properties for Parabolic and Elliptic PDE's (GPPEPDEs 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 176))

Abstract

We investigate the scale-invariant structure of the critical Hardy inequality in a unit ball under the power-type scaling. First we consider the remainder term of the critical Hardy inequality which is characterized by the ratio with or the distance from the “virtual minimizer” for the associated variational problem. We also focus on the scale invariance property of the inequality under power-type scaling and investigate the iterated scaling structure of remainder terms. Finally, we give a relation between the usual scaling enjoyed by the classical Hardy inequality and the power-type scaling via the transformation introduced by Horiuchi and Kumlin. As a by-product, we give a relationship between the Moser sequences and the Talenti functions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adimurthi, Chaudhuri, N., Ramaswamy, M.: An improved Hardy-Sobolev inequality and its application. Proc. Am. Math. Soc. 130, 489–505 (2002). (electronic)

    Google Scholar 

  2. Alvino, A.: A limit case of the Sobolev inequality in Lorentz spaces. Rend. Accad. Sci. Fis. Mat. Napoli 44, 105–112 (1977)

    MathSciNet  Google Scholar 

  3. Bianchi, G., Egnell, H.: A note on the Sobolev inequality. J. Funct. Anal. 100, 18–24 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barbatis, G., Filippas, S., Tertikas, A.: Series expansion for \(L^p\) Hardy inequalities. Indiana Univ. Math. J. 52, 171–190 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics. Academic Press (1988)

    Google Scholar 

  6. Brezis, H., Vázquez, J.L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid 10, 443–469 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Cassani, D., Ruf, B., Tarsi, C.: Best constants for Moser type inequalities in Zygmund spaces. Mat. Contemp. 36, 79–90 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Cassani, D., Ruf, B., Tarsi, C.: Group invariance and Pohozaev identity in Moser-type inequalities. Commun. Contemp. Math. 15, 1250054 (20 pages) (2013)

    Google Scholar 

  9. Cianchi, A.: A sharp embedding theorem for Orlicz-Sobolev spaces. Indiana Univ. Math. J. 45, 39–66 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cianchi, A., Ferone, A.: Hardy inequalities with non-standard remainder terms. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 889–906 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cianchi, A., Ferone, A.: Best remainder norms in Sobolev-Hardy inequalities. Indiana Univ. Math. J. 58, 1051–1096 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Detalla, A., Horiuchi, T., Ando, H.: Missing terms in Hardy-Sobolev inequalities and its application. Far East J. Math. Sci. (FJMS) 14, 333–359 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Detalla, A., Horiuchi, T., Ando, H.: Sharp remainder terms of Hardy-Sobolev inequalities. Math. J. Ibaraki Univ. 37, 39–52 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Edmunds, D.E., Kerman, R., Pick, L.: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. J. Funct. Anal. 170, 307–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Edmunds, D.E., Triebel, H.: Sharp Sobolev embeddings and related Hardy inequalities: the critical case. Math. Nachr. 207, 79–92 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Frank, R.L., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255, 3407–3430 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Filippas, S., Tertikas, A.: Optimizing improved Hardy inequalities. J. Funct. Anal. 192, 186–233 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Horiuchi, T., Kumlin, P.: On the Caffarelli-Kohn-Nirenberg type inequalities involving critical and supercritical weights. Kyoto J. Math. 52, 661–742 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ioku, N., Ishiwata, M.: A scale invariant form of a critical Hardy inequality. Int. Math. Res. Not. 18, 8830–8846 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Leray, J.: Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’Hydrodynamique. Journal de Mathématiques Pures et Appliquées 12, 1–82 (1933)

    MATH  Google Scholar 

  21. Lindqvist, P.: On the equation \({\rm {div}}(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0\). Proc. Am. Math. Soc. 109, 157–164 (1990)

    MathSciNet  MATH  Google Scholar 

  22. Machihara, S., Ozawa, T., Wadade, H.: Hardy type inequalities on balls. Tohoku Math. J. 65, 321–330 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970/71)

    Google Scholar 

  24. Peral, I., Vázquez, J.L.: On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term. Arch. Ration. Mech. Anal. 129, 201–224 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in \(\mathbb{R}^2\). J. Funct. Anal. 219, 340–367 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Science and Engineering, Ehime University, Matsuyama, Ehime, 790-8577, Japan

    Norisuke Ioku

  2. Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan

    Michinori Ishiwata

Authors
  1. Norisuke Ioku
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michinori Ishiwata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Norisuke Ioku .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Milano, Milano, Italy

    Filippo Gazzola

  2. Mathematical Institute, Tohoku University, Sendai, Japan

    Kazuhiro Ishige

  3. Dip. di Matematica e Applicazioni, Università degli Studi di Napoli Federico II, Monte S. Angelo, Napoli, Italy

    Carlo Nitsch

  4. Dipartimento di Matematica Ulisse Dini, Università degli Studi di Firenze, Firenze, Firenze, Italy

    Paolo Salani

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Ioku, N., Ishiwata, M. (2016). A Note on the Scale Invariant Structure of Critical Hardy Inequalities. In: Gazzola, F., Ishige, K., Nitsch, C., Salani, P. (eds) Geometric Properties for Parabolic and Elliptic PDE's. GPPEPDEs 2015. Springer Proceedings in Mathematics & Statistics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-41538-3_7

Download citation

Publish with us

Policies and ethics

Search

Navigation