Equivalence of Sharp Trudinger-Moser Inequalities in Lorentz-Sobolev Spaces

  • Hanli TangEmail author


The critical and subcritical Trudinger-Moser inequalities in Lorentz Sobolev space have been studied by Cassani and Tarsi (Asymptot. Anal. 64(1-2):29–51, 2009), Lu and Tang (Adv. Nonlinear Stud. 16(3):581–601, 2016). In this paper, we will prove that these critical and subcritical Trudinger-Moser inequalities are actually equivalent and thus extend those equivalence results of Lam et al. (Rev. Mat. Iberoam 33(4):1219–1246, 2017) into Lorentz Sobolev spaces.


Critical Trudinger-Moser inequality Subcritical Trudinger-Moser inequalities Lorentz-Sobolev spaces Equivalence 


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The author wishes to thank the referees very much for their very careful reading and many useful comments and suggestions on the improvement of the exposition of the paper.


  1. 1.
    Adachi, S., Tanaka, K.: Trudinger type inequalities in \(\mathbb {R}^{N}\) and their best exponents. Proc. Amer. Math. Soc. 128, 2051–2057 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adams, D.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. (2) 128(2), 385–398 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alvino, A., Ferone, V., Trombetti, G.: Moser-type inequalities in Lorentz spaces. Potential Anal. 5, 273–299 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    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)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cao, D.: Nontrivial solution of semilinear elliptic equation with critical exponent in \(\mathbb {R}^{2}\). Comm. Partial Differ. Equ. 17(3-4), 407–435 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Carleson, L., Chang, S.Y.A.: On the existence of an extremal function for an inequality of. J. Moser. Bull. Sci. Math. (2) 110(2), 113–127 (1986)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chen, L., Li, J., Lu, G., Zhang, C.: Sharpened Adams inequality and ground state solutions to the bi-Laplacian equation in R 4. Adv. Nonlinear Stud. 18(3), 429–452 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cohn, W.S., Lu, G.: Best constants for Trudinger-Moser inequalities on the Heisenberg group. Indiana Univ. Math. J. 50(4), 1567–1591 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    do Ó, J.M.: N-Laplacian equations in \(\mathbb {R}^{N}\) with critical growth. Abstr. Appl. Anal. 2(3-4), 301–315 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Flucher, M.: Extremal functions for the Trudinger-Moser inequality in 2 dimensions. Comment. Math. Helv. 67(3), 471–497 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hudson, A., Leckband, M.: A Sharp exponential inequality for Lorentz-Sobolev spaces on bounded domains. Proc. Amer. Math. Soc. 127, 2029–2033 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lam, N., Lu, G.: Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications. Adv. Math. 231, 3259–3287 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lam, N., Lu, G.: A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument. J. Differ. Equ. 255(3), 298–325 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lam, N., Lu, G., Tang, H.: Sharp subcritical Moser-Trudinger inequalities on Heisenberg groups and subelliptic PDEs. Nonlinear Anal. 95, 77–92 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lam, N., Lu, G., Zhang, L.: Equivalence of critical and subcritical sharp Moser-Trudinger-Adams inequalities. Rev. Mat. Iberoam 33(4), 1219–1246 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li, Y.: Moser-Trudinger inequality on compact Riemannian manifolds of dimension two. J. Partial Differ. Equ. 14(2), 163–192 (2001)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Li, Y.: Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds. Sci. China Ser. A 48(5), 618–648 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, J., Lu, G., Zhu, M.: Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg groups and existence of ground state solutions. Calc. Var. Partial Differential Equations 57(3), 26 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lieb, E., Loss, M.: Analysis, 2nd edn., vol. 14. Amer. Math. Soc., Providence (2001)Google Scholar
  21. 21.
    Lin, K.: Extremal functions for Moser’s inequality. Trans. Amer. Math. Soc. 348 (7), 2663–2671 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lu, G., Tang, H.: Sharp singular Trudinger-Moser inequalities in Lorentz-Sobolev spaces. Adv. Nonlinear Stud. 16(3), 581–601 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970/71)Google Scholar
  24. 24.
    Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in \(\mathbb {R}^{2}\). J. Funct. Anal. 219(2), 340–367 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    O’Neil, R.: L(p, q) spaces Convolution operators. Duke Math. J. 30, 129–142 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pohožaev, S.I.: On the Sobolev Embedding in the Case P l = N, Proceedings of the Technical Scientific Conference on Advances of Scientific Research 1964-1965. Mathematics Section, pp. 158–170. Moskov. Energet. Inst., Moscow (1965)Google Scholar
  27. 27.
    Stein, E., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, p. 297. Princeton University Press, Princeton (1971)Google Scholar
  28. 28.
    Strauss, W.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149–162 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Talenti, G.: Elliptic equations and rearrangements. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3, 697–718 (1976)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Trudinger, N.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Yudovič, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. (Russian) Dokl. Akad. Nauk SSSR 138, 805–808 (1961)MathSciNetGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal UniversityBeijingChina

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