Skip to main content

Advertisement

Log in

Canceling effects in higher-order Hardy–Sobolev inequalities

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

A classical first-order Hardy–Sobolev inequality in Euclidean domains, involving weighted norms depending on powers of the distance function from their boundary, is known to hold for every, but one, value of the power. We show that, by contrast, the missing power is admissible in a suitable counterpart for higher-order Sobolev norms. Our result complements and extends contributions by Castro and Wang (Calc Var 39(3–4):525–531, 2010), and Castro et al. (Comptes Rendus Math Acad Sci Paris 349:765–767, 2011; J Eur Math Soc 15:145–155, 2013), where a surprising canceling phenomenon underling the relevant inequalities was discovered in the special case of functions with derivatives in \(L^1\).

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, D.R.: Traces of potentials arising from translation invariant operators. Ann. Sc. Norm. Super. Pisa 25, 203–217 (1971)

    MathSciNet  MATH  Google Scholar 

  2. Adams, D.R.: A trace inequality for generalized potentials. Stud. Math. 48, 99–105 (1973)

    MathSciNet  MATH  Google Scholar 

  3. Castro, H., Wang, H.: A Hardy type inequality for \(W^{m,1}(0,1)\) functions. Calc. Var. 39(3–4), 525–531 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Castro, H., Dávila, J., Wang, H.: A Hardy type inequality for \(W^{2,1}_0(\Omega )\) functions. Comptes Rendus Math. Acad. Sci. Paris 349, 765–767 (2011)

    Article  MathSciNet  Google Scholar 

  5. Castro, H., Dávila, J., Wang, H.: A Hardy type inequality for \(W^{m,1}_0(\Omega )\) functions. J. Eur. Math. Soc. 15, 145–155 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cianchi, A., Edmunds, D.E., Gurka, P.: On weighted Poincaré inequalities. Math. Nachr. 180, 15–41 (1996)

    MathSciNet  MATH  Google Scholar 

  7. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, New York (1993)

    MATH  Google Scholar 

  8. Kufner, A.: Weighted Sobolev spaces. Teubner Verlagsgesellschaft Leipzig (1980)

  9. Horiuchi, T.: The imbedding theorems for weighted Sobolev spaces. J. Math. Kyoto Univ. 29, 365–403 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Horiuchi, T.: The imbedding theorems for weighted Sobolev spaces. II. Bull. Fac. Sci. Ibaraki Univ. Ser. A 23, 11–37 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Maz’ya, V.G.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Springer, Berlin (2011)

    MATH  Google Scholar 

  12. Murthy, M.K.V., Stampacchia, G.: Boundary value problems for some degenerate-elliptic operators. Ann. Math. Pura Appl. 80, 1–122 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  13. Opic, B., Kufner, A.: Hardy-Type Inequalities. Longman Scientific & Technical, New York (1990)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Yoshinori Yamasaki for some helpful discussions. This research was initiated during a visit of the second named author at the Department of Mathematics and Informatics “U.Dini” of the University of Florence, in the fall-winter semester 2013–2014. He wishes to thank the members of the Department for their kind hospitality. This work was partly funded by: Research project of MIUR (Italian Ministry of Education, University and Research) Prin 2012 “Elliptic and parabolic partial differential equations: geometric aspects, related inequalities, and applications” (Grant Number 2012TC7588); GNAMPA of the Italian INdAM (National Institute of High Mathematics); JSPS KAKENHI (Grant Number 25220702).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrea Cianchi.

Additional information

Communicated by H. Brezis.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cianchi, A., Ioku, N. Canceling effects in higher-order Hardy–Sobolev inequalities. Calc. Var. 56, 31 (2017). https://doi.org/10.1007/s00526-017-1112-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-017-1112-1

Mathematics Subject Classification

Navigation