Sobolev Spaces

  • Petteri Harjulehto
  • Peter Hästö
Part of the Lecture Notes in Mathematics book series (LNM, volume 2236)


In this chapter we study Sobolev spaces with generalized Orlicz integrability. We point out the novelties in this new setting and assume that the readers are familiar with classical Sobolev spaces.


  1. 2.
    R. Adams. Sobolev Spaces. Pure and Applied Mathematics, Vol. 65 (Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975)Google Scholar
  2. 3.
    R. Adams, J. Fournier, Sobolev Spaces, volume 140 of Pure and Applied Mathematics (Amsterdam), 2nd edn. (Elsevier/Academic Press, Amsterdam, 2003)Google Scholar
  3. 12.
    B. Bojarski, Remarks on Sobolev imbedding inequalities, in Complex Analysis, Joensuu 1987, volume 1351 of Lecture Notes in Math. (Springer, Berlin, 1988), pp. 52–68Google Scholar
  4. 27.
    D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis (Birkhäuser, Basel, 2013)CrossRefGoogle Scholar
  5. 30.
    D. Cruz-Uribe, J.M. Martell, C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, volume 215 of Operator Theory: Advances and Applications (Birkhäuser/Springer Basel AG, Basel, 2011)Google Scholar
  6. 34.
    L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev spaces with Variable Exponents, volume 2017 of Lecture Notes in Mathematics (Springer, Heidelberg, 2011)zbMATHGoogle Scholar
  7. 36.
    L. Diening, S. Schwarzacher, Global gradient estimates for the p(⋅)-Laplacian. Nonlinear Anal. 106, 70–85 (2014)MathSciNetCrossRefGoogle Scholar
  8. 46.
    D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics (Springer, Berlin, 2001). Reprint of the 1998 editionzbMATHGoogle Scholar
  9. 63.
    J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations (Dover Publications, Mineola, NY, 2006). Unabridged republication of the 1993 originalGoogle Scholar
  10. 88.
    J. Malý, W.P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, volume 51 of Mathematical Surveys and Monographs (American Mathematical Society, Providence, RI, 1997)CrossRefGoogle Scholar
  11. 118.
    Y.G. Reshetnyak, Integral representations of differentiable functions in domains with a nonsmooth boundary. Sibirsk. Mat. Zh. 21(6), 108–116, 221 (1980)MathSciNetCrossRefGoogle Scholar
  12. 131.
    W. Ziemer, Weakly Differentiable Functions, volume 120 of Graduate Texts in Mathematics (Springer, New York, 1989). Sobolev spaces and functions of bounded variationGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Petteri Harjulehto
    • 1
  • Peter Hästö
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

Personalised recommendations