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Positivity

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

Sobolev type inequalities for rearrangement invariant spaces

  • Guillermo P. CurberaEmail author
  • Werner J. Ricker
Article

Abstract

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.

Keywords

Sobolev inequality Rearrangement invariant space 

Mathematics Subject Classification (2000)

46E35 46E30 

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References

  1. 1.
    Adams R.A., Fournier J.J.F.: Sobolev Spaces. Academic Press Inc., Boston (2003)zbMATHGoogle Scholar
  2. 2.
    Bennett C., Sharpley R.: Interpolation of Operators. Academic Press Inc., Boston (1988)zbMATHGoogle Scholar
  3. 3.
    Cianchi A., Pick L.: Sobolev embeddings into BMO, VMO, and L spaces. Ark. Mat. 36, 317–340 (1998)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Edmunds D., Kerman R., Pick L.: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. J. Funct. Anal. 170, 307–355 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ferone A., Volpicelli R.: Polar factorization and pseudo-rearrangements: applications to Pólya–Szegö type inequalities. Nonlinear Anal. 53, 929–949 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kerman R., Pick L.: Optimal Sobolev imbeddings. Forum Math. 18, 535–570 (2006)MathSciNetzbMATHCrossRefGoogle 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

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