Revista Matemática Complutense

, Volume 25, Issue 1, pp 247–265 | Cite as

On dimension-free Sobolev imbeddings II

  • Miroslav KrbecEmail author
  • Hans-Jürgen Schmeisser


We prove dimension-invariant imbedding theorems for Sobolev spaces using extrapolation means.


Sobolev space Imbedding theorem Uncertainty principle Best constants for imbeddings 

Mathematics Subject Classification (2000)

46E35 46E30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, D.: Traces of potentials arising from translation invariant operators. Ann. Sc. Norm. Super. Pisa. 25, 1–9 (1971) Google Scholar
  2. 2.
    Adams, R.A. Sobolev Spaces. Academic Press, New York (1975) zbMATHGoogle Scholar
  3. 3.
    Alvino, A.: Sulla diseguaglianza di Sobolev in spazi di Lorentz. Boll. Unione Mat. Ital. 5(14-A), 148–156 (1977) MathSciNetGoogle Scholar
  4. 4.
    Bennett, C., Rudnick, K.: On Lorentz-Zygmund spaces. Diss. Math. CLXXV, 1–72 (1980) Google Scholar
  5. 5.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988) zbMATHGoogle Scholar
  6. 6.
    Chiarenza, F., Frasca, M.: A remark on a paper by C. Fefferman. Proc. Am. Math. Soc. 108, 407–409 (1990) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Cruz-Uribe, D., Krbec, M.: Localization and extrapolation in Lorentz-Orlicz spaces. In: Cwikel, M. et al. (eds.) Function Spaces, Interpolation Theory and Related Topics. Proc. Conf. Lund (Sweden), 17–22 August 2001, pp. 389–401. de Gruyter, Berlin (2002) Google Scholar
  8. 8.
    Edmunds, D.E., Edmunds, R.M., Triebel, H.: Entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces. J. Lond. Math. Soc. 2(35), 121–134 (1987) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Fefferman, C.: The uncertainty principle. Bull. Am. Math. Soc. 9, 129–206 (1983) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Ishii, J.: On equivalence of modular function spaces. Proc. Jpn. Acad. Sci. 35, 551–556 (1959) CrossRefzbMATHGoogle Scholar
  11. 11.
    Iwaniec, T., Verde, A.: On the operator L(f)=flog |f|. J. Funct. Anal. 169, 391–420 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Kerman, R., Sawyer, E.: The trace inequality and eigenvalue estimates for Schrödinger operators. Ann. Inst. Fourier 36, 207–228 (1986) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Krasnosel’skii, M.A., Rutitskii, Ya.B.: Convex Functions and Orlicz Spaces. Noordhoff, Amsterdam (1961) Google Scholar
  14. 14.
    Krbec, M., Schmeisser, H.-J.: A limiting case of the uncertainty principle. In: Fila, M. et al. (eds.) Proceedings of Equadiff 11, Proceedings of Minisymposia and Contributed Talks Bratislava, 25–29 July 2005, pp. 181–187 (2007) Google Scholar
  15. 15.
    Krbec, M., Schmeisser, H.-J.: Dimension-free imbeddings of Sobolev spaces. Preprint, Prague (2008) Google Scholar
  16. 16.
    Krbec, M., Schmeisser, H.-J.: On dimension-free Sobolev imbeddings I. To appear Google Scholar
  17. 17.
    Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14. Am. Math. Soc., Providence (2001) zbMATHGoogle Scholar
  18. 18.
    Martín, J., Milman, M.: Pointwise symmetrization inequalities for Sobolev functions and applications. Adv. Math. 225, 121–199 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Maz’ya, V.G.: Classes of domains and embedding theorems for functional spaces. Dokl. Akad. Nauk SSSR 133, 527–530 (1960) Google Scholar
  20. 20.
    Maz’ya, V.G.: On the theory of the n-dimensional Schrödinger operator. Izv. Akad. Nauk SSSR, Ser. Mat. 28, 1145–1172 (1964) zbMATHMathSciNetGoogle Scholar
  21. 21.
    Milman, M.: Extrapolation and Optimal Decompositions. Springer, Berlin (1994) zbMATHGoogle Scholar
  22. 22.
    Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Math., vol. 1034. Springer, Berlin (1983) zbMATHGoogle Scholar
  23. 23.
    Schmeisser, H.-J., Sickel, W.: Traces, Gagliardo-Nirenberg inequalities and Sobolev type embeddings for vector-valued function spaces. Jenaer Schriften zur Mathematik und Informatik, Math/Inf/24/01, Jena (2001) Google Scholar
  24. 24.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976) CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Triebel, H.: Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces. Proc. Lond. Math. Soc. 3(66), 589–618 (1993) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Triebel, H.: Theory of Function Spaces III. Birkhäuser, Basel (2006) zbMATHGoogle Scholar
  27. 27.
    Triebel, H.: Tractable embeddings of Besov spaces into Zygmund spaces. Preprint, Jena (2009) Google Scholar

Copyright information

© Revista Matemática Complutense 2011

Authors and Affiliations

  1. 1.Institute of MathematicsAcademy of Sciences of the Czech RepublicPraha 1Czech Republic
  2. 2.Mathematisches Institut, Fakultät für Mathematik und InformatikFriedrich-Schiller-UniversitätJenaGermany

Personalised recommendations