Skip to main content
SpringerLink
Log in

Sharp Second-Order Adams Inequalities in Lorentz–Sobolev Spaces

  • Original Research Article
  • Published:
La Matematica Aims and scope Submit manuscript

Abstract

In this paper, we establish sharp subcritical and critical second-order Adams inequalities in Lorentz–Sobolev spaces. We also prove the subcritical and critical Adams inequalities are actually equivalent and our results extend existing ones in Tang (Potential Anal 53(1):297–314, 2020) to second order.

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

Access this article

Log in via an institution

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

Data availability

Data will be made available on request.

References

  1. Adachi, S., Tanaka, K.: Trudinger type inequalities in \(R^{N}\) and their best exponents. Proc. of the Amer. Math. Soc. 128, 2051–2057 (1999)

    Article  MATH  Google Scholar 

  2. Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. of Math 128(2), 385–398 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alberico, A.: Moser type inequalities for higher-order derivatives in Lorentz spaces. Potential Anal. 28(4), 389–400 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alvino, A., Ferone, V., Trombetti, G.: Moser-type inequalities in Lorentz spaces. Potential Anal. 5, 273–299 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carleson, L., Chang, S. Y. A.: On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. 110(2), 113–127 (1986)

  6. Cassani, D., Tarsi, C.: A Moser-type inequality in Lorentz-Sobolev spaces for unbounded domains in \({\mathbb{R} }^{N}\). Asymptot. Anal. 64(1–2), 29–51 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Chen, L., Lu, G., Zhu, M.: Existence and nonexistence of extremals for critical Adams inequalities in\(\mathbb{R}^{4}\)and Trudinger-Moser inequalities in\(\mathbb{R}^{2}\), Adv. Math. 368, 107143, 61 (2020)

  8. Flucher, M.: Extremal functions for the Trudinger-Moser inequality in 2 dimensions. Comment. Math. Helv. 67(3), 471–497 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fontana, L., Morpurgo, C.: Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on \({\mathbb{R} }^{n}\). Nonlinear Anal. 167, 85–122 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hudson, A., Leckband, M.: A Sharp exponential inequality for Lorentz-Sobolev spaces on bounded domains. Proc. Amer. Math. Soc. 127, 2029–2033 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ibrahim, S., Masmoudi, N., Nakanishi, K.: Trudinger-Moser inequality on the whole plane with the exact growth condition. J. Eur. Math. Soc. (JEMS) 17(4), 819–835 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jr. Jodeit, M.: An inequality for the indefinite integral of a function in Lq, in Collection of Articles Honoring the Completion by Antoni Zygmund of 50 Years of Scientific Activity, VI. Studia Mathematica, Vol. 44 (Instytut Matematyczny Polskiej Akademii Nauk, 1972), 545–554

  13. Lam, N., Lu, G.: Sharp Adams type inequalities in Sobolev spaces \(W^{m,\frac{m}{n}}({\mathbb{R} }^n)\)for arbitrary integer \(m\). J. Differential Equations 253, 1143–1171 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lam, N., Lu, G., Zhang, L.: Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities. Rev. Mat. Iberoam. 33(4), 1219–1246 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, Y.: Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds. Sci. China Ser. A 48(5), 618–648 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, Y., Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in \( {\mathbb{R} }^{n}\). Indiana Univ. Math. J. 57(1), 451–480 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lieb, E., Loss, M.: Analysis, 2nd edu, Gradute Studies in Mathematics, vol. 14. Amer. Math. Soc, Providence, RI (2001)

    Google Scholar 

  18. Lin, K.: Extremal functions for Moser’s inequality. Trans. Amer. Math. Soc. 348(7), 2663–2671 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lu, G., Tang, H.: Sharp singular Trudinger-Moser inequalities in Lorentz-Sobolev spaces. Adv. Nonlinear Stud. 16(3), 581–601 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lu, G., Tang, H., Zhu, M.: Best constants for Adams’ inequalities with the exact growth condition in \({\mathbb{R} }^n\). Adv. Nonlinear Stud. 15(4), 763–788 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Masmoudi, N., Sani, F.: Adams’ inequality with the exact growth condition in \({\mathbb{R} }^4\). Comm. Pure Appl. Math. 67(8), 1307–1335 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. : Masmoudi, N., Sani, F.: Higher order Adams’ inequality with the exact growth condition, Commun. Contemp. Math. 20(6)(2018), 1750072, 33 pp

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

  24. Pohožaev, S.I.: On the eigenfunctions of the equation \(\Delta u+\lambda f(u)=0\), (Russian) Dokl. Akad. Nauk SSSR 165, 36–39 (1965)

    MathSciNet  Google Scholar 

  25. Stein, E., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971. 297 pp

  26. Strauss, W.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149–162 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  27. Talenti, G.: Elliptic equations and rearrangements. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3, 697–718 (1976)

    MathSciNet  MATH  Google Scholar 

  28. Tang, H.: Equivalence of sharp Trudinger-Moser inequalities in Lorentz-Sobolev spaces. Potential Anal. 53(1), 297–314 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)

    MathSciNet  MATH  Google Scholar 

  30. Yudovič, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations, (Russian) Dokl. Akad. Nauk SSSR 138, 805–808 (1961)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Hanli Tang

Authors
  1. Hanli Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hanli Tang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author was supported by National Key Research and Development Program (No. 2020YFA0712900).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, H. Sharp Second-Order Adams Inequalities in Lorentz–Sobolev Spaces. La Matematica 2, 772–787 (2023). https://doi.org/10.1007/s44007-023-00069-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s44007-023-00069-8

Keywords

Search

Navigation