The Journal of Geometric Analysis

, Volume 26, Issue 4, pp 2930–2954 | Cite as

Optimal Rearrangement Invariant Sobolev Embeddings in Mixed Norm Spaces

  • Nadia Clavero
  • Javier Soria


We improve the Sobolev-type embeddings due to Gagliardo (Ric Mat 7:102–137, 1958) and Nirenberg (Ann Sc Norm Sup Pisa 13:115–162, 1959) in the setting of rearrangement invariant (r.i.) spaces. In particular, we concentrate on seeking the optimal domains and the optimal ranges for these embeddings between r.i. spaces and mixed norm spaces. As a consequence, we prove that the classical estimate for the standard Sobolev space \(W^{1}L^{p}\) by Poornima (Bull Sci Math 107(3):253–259,  1983), O’Neil (Duke Math J 30:129–142,  1963) and Peetre (Ann Inst Fourier 16(1):279–317,  1966) (\(1 \le p < n\)), and by Hansson (Math Scand 45(1):77–102,  1979, Brezis and Wainger (Commun Partial Differ Equ 5(7):773–789,  1980) and Maz’ya (Sobolev spaces,  1985) (\(p=n\)) can be further strengthened by considering mixed norms on the target spaces.


Sobolev embeddings Rearrangement-invariant spaces Hardy operator Optimal range Optimal domain 

Mathematics Subject Classification

28A35 46E30 46E35 



We would like to thank the referee for his/her careful revision which has improved the final version of this work. Both authors have been partially supported by the Grants MTM2013-40985-P (Spanish Government) and 2014SGR289 (Catalan Autonomous Government).


  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), vol. 140, 2nd edn. Elsevier/Academic Press, Amsterdam (2003)Google Scholar
  2. 2.
    Algervik, R., Kolyada, V.I.: On Fournier-Gagliardo mixed norm spaces. Ann. Acad. Sci. Fenn. Math. 36(2), 493–508 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Barza, S., Kamińska, A., Persson, L., Soria, J.: Mixed norm and multidimensional Lorentz spaces. Positivity 10(3), 539–554 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Benedek, A., Panzone, R.: The space \(L^{p}\), with mixed norm. Duke Math. J. 28, 301–324 (1961)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bennett, C., Rudnick, K.: On Lorentz-Zygmund spaces. Diss. Math. 175, 1–72 (1980)MathSciNetMATHGoogle Scholar
  6. 6.
    Bennett, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics, vol. 129. Academic Press Inc., Boston (1988)MATHGoogle Scholar
  7. 7.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976)CrossRefMATHGoogle Scholar
  8. 8.
    Blei, R.C., Fournier, J.J.F.: Mixed-norm conditions and Lorentz norms. Commut. Harmon. Anal. 91, 57–78 (1989)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Blozinski, A.P.: Multivariate rearrangements and Banach function spaces with mixed norms. Trans. Am. Math. Soc. 263, 149–167 (1981)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Boccuto, A., Bukhvalov, A.V., Sambucini, A.R.: Some inequalities in classical spaces with mixed norms. Positivity 6, 393–411 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Brezis, H.: Analyse Fonctionnelle. Collection Mathématiques Appliqués pour la Maîtrise. Masson, Paris (1983)Google Scholar
  12. 12.
    Brezis, H., Wainger, S.: A note on limiting cases of Sobolev embeddings and convolution inequalities. Commun. Partial Differ. Equ. 5(7), 773–789 (1980)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bukhvalov, A.V.: Spaces with mixed norm, Vestnik Leningrad. Univ. no. 19. Mat. Meh. Astronom. Vyp. 4, 5–12, 151 (1973)Google Scholar
  14. 14.
    Carro, M.J., Pick, L., Soria, J., Stepanov, V.D.: On embeddings between classical Lorentz spaces. Math. Inequal. Appl. 4, 397–428 (2001)MathSciNetMATHGoogle Scholar
  15. 15.
    Cianchi, A.: Symmetrization and second-order Sobolev inequalities. Ann. Mat. Pura Appl. (4) 183(1), 45–77 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Clavero, N., Soria, J.: Mixed norm spaces and rearrangement invariant estimates. J. Math. Anal. Appl. 419, 878–903 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    DeVore, R., Scherer, K.: Interpolation of linear operators on Sobolev spaces. Ann. Math. 109, 583–599 (1979)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Edmunds, D.E., Kerman, R., Pick, L.: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. J. Funct. Anal. 170, 307–355 (2000)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fournier, J.J.F.: Mixed norms and rearrangements: Sobolev’s inequality and Littlewood’s inequality. Ann. Mat. Pura Appl. 148, 51–76 (1987)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gagliardo, E.: Proprietà di alcune classi di funzioni in più variabili. Ric. Mat. 7, 102–137 (1958)MathSciNetMATHGoogle Scholar
  21. 21.
    Hansson, K.: Imbedding theorems of Sobolev type in potential theory. Math. Scand. 45(1), 77–102 (1979)MathSciNetMATHGoogle Scholar
  22. 22.
    Holmstedt, T.: Interpolation of quasi-normed spaces. Math. Scand. 26, 177–199 (1970)MathSciNetMATHGoogle Scholar
  23. 23.
    Kerman, R., Pick, L.: Optimal Sobolev imbeddings. Forum Math. 18(4), 535–570 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kolyada, V.I.: Iterated rearrangements and Gagliardo-Sobolev type inequalities. J. Math. Anal. Appl. 387, 335–348 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Kolyada, V.I.: On Fubini type property in Lorentz spaces. Recent Adv. Harmonic Anal. Appl. 25, 171–179 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Maz’ya, V.: Sobolev Spaces, Springer Series in Soviet Mathematics. Springer, Berlin (1985)Google Scholar
  27. 27.
    Milman, M.: Notes on interpolation of mixed norm spaces and applications. Q. J. Math. Oxf. 42(167), 325–334 (1991)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Nirenberg, L.: On elliptic partial differential equations. Ann. Sc. Norm. Sup. Pisa 13, 115–162 (1959)MathSciNetMATHGoogle Scholar
  29. 29.
    O’Neil, R.: Convolution operators and \(L(p, q)\) spaces. Duke Math. J. 30, 129–142 (1963)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Peetre, J.: Espaces d’interpolation et théorème de Soboleff. Ann. Inst. Fourier 16(1), 279–317 (1966)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Poornima, S.: An embedding theorem for the Sobolev space \(W^{1,1}\). Bull. Sci. Math. 107(3), 253–259 (1983)MathSciNetMATHGoogle Scholar
  32. 32.
    Sobolev, S.L.: On a theorem of functional analysis, Math. Sb. 46, 471–496 (1963), translated in Am. Math. Soc. Transl. 34, 39–68 (1938)Google Scholar
  33. 33.
    Talenti, G.: Inequalities in rearrangement invariant function spaces, nonlinear analysis, function spaces and applications, (Prague, 1994), vol. 5. Prometheus, Prague (1994)MATHGoogle Scholar

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© Mathematica Josephina, Inc. 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics and AnalysisUniversity of BarcelonaBarcelonaSpain

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