Arkiv för Matematik

, Volume 43, Issue 1, pp 1–28 | Cite as

Differentiability properties of Orlicz-Sobolev functions

  • Angela Alberico
  • Andrea Cianchi
Article

Keywords

Differentiability Property 

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References

  1. [A1]
    Adams, D. R., Lectures onL p-potential theory,Umeå Univ. Reports 2 (1981).Google Scholar
  2. [A2]
    Adams, D. R., Choquet integrals in potential theory,Publ. Mat. 42 (1998), 3–66.MATHMathSciNetGoogle Scholar
  3. [AH]
    Adams, D. R. andHedberg, L. I.,Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996.Google Scholar
  4. [AHS1]
    Adams, D. R. andHurri-Syrjänen, R., Capacity estimates,Proc. Amer. Math. Soc. 131 (2003), 1159–1167.CrossRefMathSciNetGoogle Scholar
  5. [AHS2]
    Adams, D. R. andHurri-Syrjänen, R., Vanishing exponential integrability for functions whose gradients belong toL n (log(e+L))α,J. Funct. Anal. 197 (2003), 162–178.CrossRefMathSciNetGoogle Scholar
  6. [AM]
    Adams, D. R. andMeyers, N. G., Thinness and Wiener criteria for non-linear potentials,Indiana Univ. Math. J. 22 (1972/73), 169–197.CrossRefMathSciNetGoogle Scholar
  7. [AB]
    Aïssaoui, N. andBenkirane, A., Capacités dans les espaces d'Orlicz,Ann. Sci. Math. Québec 18 (1994), 1–23.Google Scholar
  8. [AFP]
    Ambrosio, L., Fusco, N. andPallara, D.,Functions of Bounded Variation and Free Discontinuity Problems, Oxford Univ. Press, New York, NY, 2000.Google Scholar
  9. [BaZ]
    Bagby, T. andZiemer, W. P., Pointwise differentiability and absolute continuity,Trans. Amer. Math. Soc. 191 (1974), 129–148.MathSciNetGoogle Scholar
  10. [BS]
    Bennett, C. andSharpley, R.,Interpolation of Operators, Academic Press, Boston, MA, 1988.Google Scholar
  11. [BW]
    Brezis, H. andWainger, S., A note on limiting cases of Sobolev embeddings and convolution inequalities,Comm. Partial Differential Equations 5 (1980), 773–789.MathSciNetGoogle Scholar
  12. [BZ]
    Brothers, J. E. andZiemer, W. P., Minimal rearrangements of Sobolev functions,J. Reine Angew. Math.,384 (1988), 153–179.MathSciNetGoogle Scholar
  13. [CFR]
    Calderón, C., Fabes, E. B. andRiviere, N. M., Maximal smoothing operators,Indiana Univ. Math. J. 23 (1973/74), 889–898.MathSciNetGoogle Scholar
  14. [C1]
    Cianchi, A., Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems,Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 576–608.MathSciNetGoogle Scholar
  15. [C2]
    Cianchi, A., A sharp embedding theorem for Orlicz-Sobolev spaces,Indiana Univ. Math. J. 45 (1996), 39–65.CrossRefMATHMathSciNetGoogle Scholar
  16. [C3]
    Cianchi, A., A fully anisotropic Sobolev inequality,Pacific. J. Math. 196 (2000), 283–295.MATHMathSciNetGoogle Scholar
  17. [C4]
    Cianchi, A., Optimal Orlicz-Sobolev embeddings,Rev. Mat. Iberoamericana 20 (2004), 427–474.MATHMathSciNetGoogle Scholar
  18. [CS]
    Cianchi, A. andStroffolini, B., An extension of Hedberg's convolution inequality and applications,J. Math. Anal. Appl. 227 (1998), 166–186.CrossRefMathSciNetGoogle Scholar
  19. [CP]
    Cwikel, M. andPustylnik, E., Sobolev type embeddings in the limiting case,J. Fourier Anal. Appl. 4 (1998), 433–446.MathSciNetGoogle Scholar
  20. [EGO]
    Edmunds, D. E., Gurka, P. andOpic, P. B., Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces,Indiana Univ. Math. J. 44 (1995), 19–43.CrossRefMathSciNetGoogle Scholar
  21. [EKP]
    Edmunds, D. E., Kerman, R. andPick, L., Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms,J. Funct. Anal. 170 (2000), 307–355.CrossRefMathSciNetGoogle Scholar
  22. [EG]
    Evans, L. C. andGariepy, R. F.,Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.Google Scholar
  23. [FZ]
    Federer, H. andZiemer, W. P., The Lebesgue set of a function whose distribution derivatives arep-th power summable,Indiana Univ. Math. J. 22 (1972/73), 139–158.CrossRefMathSciNetGoogle Scholar
  24. [FLS]
    Fusco, N., Lions, P.-L. andSbordone, C., Sobolev imbedding theorems in borderline cases,Proc. Amer. Math. Soc. 124 (1996), 561–565.CrossRefMathSciNetGoogle Scholar
  25. [H]
    Hansson, K., Imbedding theorems of Sobolev type in potential theory,Math. Scand. 45 (1979), 77–102.MATHMathSciNetGoogle Scholar
  26. [He]
    Hedberg, L. I., Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem,Acta Math. 147 (1981), 237–264.MATHMathSciNetGoogle Scholar
  27. [KKM]
    Kauhanen, J., Koskela, P. andMalý, J., On functions with derivatives in a Lorentz space,Manuscripta Math. 100 (1999), 87–101.CrossRefMathSciNetGoogle Scholar
  28. [MSZ]
    Malý, J., Swanson, D. andZiemer, W. P., Fine behaviour of functions with gradients in a Lorentz space,In preparation.Google Scholar
  29. [MZ]
    Malý, J. andZiemer, W. P.,Fine Regularity of Solutions of Elliptic Partial Differential Equations, math. Surveys Monogr.51, Amer. Math. Soc., Providence, RI, 1997.Google Scholar
  30. [M]
    Maźya, V. G.,Sobolev Spaces, Springer-Verlag, Berlin, 1985.Google Scholar
  31. [Me]
    Meyers, N. G., Taylor expansion of Bessel potentials,Indiana Univ. Math. J. 23 (1973/74), 1043–1049.MathSciNetGoogle Scholar
  32. [M1]
    Mizuta, Y., Fine differentiability of Riesz potentials,Hiroshima Math. J. 8 (1978), 505–514.MATHMathSciNetGoogle Scholar
  33. [M2]
    Mizuta, Y.,Potential Theory in Euclidean spaces, GAKUTO Internat. Series Mathematical Sciences and Applications,6, Gakkotosho, Tokyo, 1996.Google Scholar
  34. [R]
    Rademacher, H., Über partielle und totale Differenzierbarkeit, I,Math. Ann. 79 (1919), 340–359.CrossRefMathSciNetGoogle Scholar
  35. [S1]
    Stein, E. M.,Singular Integrals and Differentiablity Properties of Functions, Princeton University Press, Princeton, NJ, 1970.Google Scholar
  36. [S2]
    Stein, E. M., Editor's note: the differentiability of functions inR n,Ann. of Math. 113 (1981), 383–385.MATHMathSciNetGoogle Scholar
  37. [T1]
    Talenti, G., An embedding theorem, inPartial Differential Equations and the Calculus of Variations, Vol. II, Progr. Nonlinear Differential Equations Appl.2, pp. 919–924, Birkhäuser, Boston, MA, 1989.Google Scholar
  38. [T2]
    Talenti, G., Boundedeness of minimizers,Hokhaido Math. J. 19 (1990), 259–279.MATHMathSciNetGoogle Scholar
  39. [Tr]
    Trudinger, N. S., On imbeddings into Orlicz spaces and some applications,J. Math. Mech. 17 (1967), 473–483.MATHMathSciNetGoogle Scholar
  40. [Z]
    Ziemer, W. P.,Weakly Differentiable Functions, Springer, New York, NY, 1989.Google Scholar

Copyright information

© Institut Mittag-Leffler 2005

Authors and Affiliations

  • Angela Alberico
    • 1
  • Andrea Cianchi
    • 2
  1. 1.Istituto per le Applicazioni del Calcolo “M. Picone”Sez. Napoli-C.N.R.NapoliItaly
  2. 2.Dipartimento di Matematica e Applicazioni per l'ArchitetturaUniversità di FirenzeFirenzeItaly

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