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Potential Analysis

, Volume 32, Issue 3, pp 201–228 | Cite as

Optimal Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving k-Modulus of Smoothness

  • Amiran Gogatishvili
  • Júlio S. Neves
  • Bohumír Opic
Article

Abstract

We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H σ X(IR n ) with order of smoothness σ ∈ (0, n), modelled upon rearrangement invariant Banach function spaces X(IR n ), into generalized Hölder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IR n ) is the Lorentz-Karamata space \(L_{p,q;b}({{\rm I\kern-.17em R}}^n)\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \(H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n)\) into generalized Hölder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield new and sharp embeddings of Sobolev-Orlicz spaces W k + 1 L n/k(logL) α (IR n ) and W k L n/k(logL) α (IR n ) into generalized Hölder spaces.

Keywords

Slowly varying functions Lorentz-Karamata spaces Rearrangement-invariant Banach function spaces Bessel potentials (fractional) Sobolev-type spaces Hölder-type spaces Zygmund-type spaces Embedding theorems 

Mathematics Subject Classifications (2000)

46E35 46E30 26B35 26A12 26D15 26A15 26A16 26B35 47B38 26D10 

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References

  1. 1.
    Adams, R.A., Fournier, J.J.: Sobolev spaces, vol. 140. Academic, Amsterdam (2003)Google Scholar
  2. 2.
    Aronszajn, N., Mulla, F., Szeptycki, P.: On spaces of potentials connected with L p classes, Part I. Ann. Inst. Fourier 13(2), 211–306 (1963)MATHMathSciNetGoogle Scholar
  3. 3.
    Aronszajn, N., Smith, K.: Theory of Bessel potentials, Part I. Ann. Inst. Fourier 11, 385–475 (1961)MATHMathSciNetGoogle Scholar
  4. 4.
    Bennett, C., Rudnick, K.: On Lorentz-Zygmund spaces. Dissertationes Math. (Rozprawy Mat.) 175, 1–72 (1980)MathSciNetGoogle Scholar
  5. 5.
    Bennett, C., Sharpley, R.: Interpolation of operators, Pure and Applied Mathematics, vol. 129. Academic, New York (1988)Google Scholar
  6. 6.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)MATHGoogle Scholar
  7. 7.
    Brézis, H., Wainger, S.: A note on limiting cases of Sobolev embeddings. Comm. Partial Differential Equations 5, 773–789 (1980)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Calderón, A.P.: Lebesgue spaces of differentiable functions and distributions. In: Partial Differential Equations, Proc. Sympos. Pure Math. 4, pp. 33–49. Amer. Math. Soc., Providence (1961)Google Scholar
  9. 9.
    Cianchi, A.: Some results in the theory of Orlicz spaces and application to variational problems. In: Krbec, M., Kufner, A. (eds.) Nonlinear Analysis, Function Spaces and Aplications, Proceedings, vol. 6, pp. 50–92. MI AS CR, Prague (1999)Google Scholar
  10. 10.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation, Grundlehren der mathematischen Wissenschaften—A series of Comprehensive Studies in Mathematics, vol. 303. Springer, Berlin (1993)Google Scholar
  11. 11.
    DeVore, R.A., Sharpley, R.C.: On the differentiability of functions in R n. Proc. Amer. Math. Soc. 91, 326–328 (1984)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Edmunds, D.E., Evans, W.D.: Hardy Operators, Function Spaces And Embeddings. Springer, Berlin (2004)MATHGoogle Scholar
  13. 13.
    Edmunds, D.E., Gurka, P., Opic, B.: Double exponential integrability, Bessel potentials and embedding theorems. Studia Math. 115, 151–181 (1995)MATHMathSciNetGoogle Scholar
  14. 14.
    Edmunds, D.E., Gurka, P., Opic, B.: On embeddings of logarithmic Bessel potential spaces. J. Funct. Anal. 146(1), 116–150 (1997)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Edmunds, D.E., Gurka, P., Opic, B.: Optimality of embeddings of logarithmic Bessel potential spaces. Q. J. Math. 51, 185–209 (2000)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Edmunds, D.E., Kerman, R., Pick, L.: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. J. Funct. Anal. 170, 307–355 (2000)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Evans, W.D., Opic, B.: Real interpolation with logarithmic functors and reiteration. Canad. J. Math. 52, 920–960 (2000)MATHMathSciNetGoogle Scholar
  18. 18.
    Gogatishvili, A., Neves, J.S., Opic, B.: Optimality of embeddings of Bessel-potential-type spaces into Lorentz-Karamata spaces. Proc. R. Soc. Edinb. 134A, 1127–1147 (2004)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Gogatishvili, A., Neves, J.S., Opic, B.: Optimality of embeddings of Bessel-potential-type spaces into generalized Hölder spaces. Publ. Mat. 49, 297–327 (2005)MATHMathSciNetGoogle Scholar
  20. 20.
    Gogatishvili, A., Neves, J.S., Opic, B.: Sharp estimates of the k-modulus of smoothness of Bessel potentials. Preprint no. 08-30. Departamento de Matemática da Universidade de Coimbra, Portugal (2008)Google Scholar
  21. 21.
    Gogatishvili, A., Neves, J.S., Opic, B.: Optimal embeddings and compact embeddings of Bessel-potential-type spaces. Math. Z. 262(3), 645–682 (2009)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Gogatishvili, A., Opic, B., Trebels, W.: Limiting reiteration for real interpolation with slowly varying functions. Math. Nachr. 278, 86–107 (2005)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Gurka, P., Opic, B.: Sharp embeddings of Besov-type spaces. J. Comput. Appl. Math. 2008, 235–269 (2007)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Heinig, H.P., Stepanov, V.D.: Weighted Hardy inequalities for increasing functions. Canad. J. Math. 45, 104–116 (1993)MATHMathSciNetGoogle Scholar
  25. 25.
    Kufner, A., John, O., Fučík, S.: Function Spaces. Academia, Prague (1977)MATHGoogle Scholar
  26. 26.
    Marić, V.: Regular variation and differential equations. Lecture Notes in Mathematics, vol. 1726. Springer, Berlin (2000)Google Scholar
  27. 27.
    Maz’ja, V.G.: Sobolev Spaces. Springer, Berlin (1985)MATHGoogle Scholar
  28. 28.
    Neves, J.S.: Fractional Sobolev type spaces and embeddings. Ph.D. thesis, University of Sussex (2001)Google Scholar
  29. 29.
    Neves, J.S.: Lorentz-Karamata spaces, Bessel and Riesz potentials and embeddings. Dissertationes Math. (Rozprawy Mat.) 405, 1–46 (2002)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Opic, B.: Embeddings of Bessel-potential-type spaces. Preprint no. 169/2007. Institute of Mathematics, AS CR, Prague (2007)Google Scholar
  31. 31.
    Opic, B., Kufner, A.: Hardy-type inequalities. Pitman Research Notes in Math. Series 219, Longman Sci. & Tech., Harlow (1990)MATHGoogle Scholar
  32. 32.
    Opic, B., Pick, L.: On generalized Lorentz-Zygmund spaces. Math. Inequal. Appl. 2(3), 391–467 (1999)MATHMathSciNetGoogle Scholar
  33. 33.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)MATHGoogle Scholar
  34. 34.
    Stepanov, V.D.: The weighted Hardy’s inequality for nonincreasing functions. Trans. Amer. Math. Soc. 338, 173–186 (1993)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Ziemer, W.P.: Weakly Differentiable Functions, vol. 120. Graduate Texts in Mathematics. Springer, Berlin (1989)Google Scholar
  36. 36.
    Zygmund, A.: Trigonometric Series, vol. I. Cambridge University Press, Cambridge (1957)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Amiran Gogatishvili
    • 1
  • Júlio S. Neves
    • 2
  • Bohumír Opic
    • 1
    • 3
  1. 1.Institute of MathematicsAcademy of Sciences of the Czech RepublicPrague 1Czech Republic
  2. 2.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  3. 3.Department of Mathematics and Didactics of MathematicsTechnical University of LiberecLiberecCzech Republic

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