, Volume 15, Issue 4, pp 553–570 | Cite as

Sobolev type inequalities for rearrangement invariant spaces



In the setting of rearrangement invariant spaces, optimal Sobolev inequalities (via the gradient) are well understood. By means of an alternative functional, we obtain new Sobolev inequalities which are finer than (and not necessarily equivalent to) the ones mentioned above.


Sobolev inequality Rearrangement invariant space 

Mathematics Subject Classification (2000)

46E35 46E30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams R.A., Fournier J.J.F.: Sobolev Spaces. Academic Press Inc., Boston (2003)MATHGoogle Scholar
  2. 2.
    Bennett C., Sharpley R.: Interpolation of Operators. Academic Press Inc., Boston (1988)MATHGoogle Scholar
  3. 3.
    Cianchi A., Pick L.: Sobolev embeddings into BMO, VMO, and L spaces. Ark. Mat. 36, 317–340 (1998)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Curbera, G.P., Ricker, W.J.: Optimal domains for the kernel operator associated with Sobolev’s inequality, Studia Math. 158, 131–152 (2003) and 170, 217–218 (2005)Google Scholar
  5. 5.
    Curbera G.P., Ricker W.J.: Compactness properties of Sobolev imbeddings for rearrangement invariant norms. Trans. Am. Math. Soc. 359, 1471–1484 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Curbera G.P., Ricker W.J.: Can optimal rearrangement invariant Sobolev imbeddings be further extended?. Indiana Univ. Math. J. 56, 1479–1497 (2007)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Edmunds D., Kerman R., Pick L.: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. J. Funct. Anal. 170, 307–355 (2000)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ferone A., Volpicelli R.: Polar factorization and pseudo-rearrangements: applications to Pólya–Szegö type inequalities. Nonlinear Anal. 53, 929–949 (2003)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kerman R., Pick L.: Optimal Sobolev imbeddings. Forum Math. 18, 535–570 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Mossino J., Temam R.: Directional derivative of the increasing rearranngement mapping and applications to a queer differential equation in plasma physics. Duke Math. J. 48, 474–495 (1981)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Talenti G.: Inequalities in rearrangements invariant function spaces. In: Krbec, M. (eds) Nonlinear Analysis, Function Spaces and Applications, vol. 5, pp. 177–230. Prometheus, Prague (1995)Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Facultad de MatemáticasUniversidad de SevillaSevilleSpain
  2. 2.Math.-Geogr. FakultätKatholische Universität Eichstätt-IngolstadtEichstättGermany

Personalised recommendations