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Revista Matemática Complutense

, Volume 27, Issue 2, pp 657–676 | Cite as

Interpolation in variable exponent spaces

  • Alexandre AlmeidaEmail author
  • Peter Hästö
Article

Abstract

In this paper we study both real and complex interpolation in the recently introduced scales of variable exponent Besov and Triebel–Lizorkin spaces. We also take advantage of some interpolation results to study a trace property and some pseudodifferential operators acting in the variable index Besov scale.

Keywords

Non-standard growth Variable exponent Besov space  Triebel–Lizorkin space Real interpolation Complex interpolation Trace operator Pseudodifferential operators 

Mathematics Subject Classification (2000)

46E35 46E30 42B15 42B25 

Notes

Acknowledgments

The first named author was supported in part by Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, Portugal, through FCT—Portuguese Foundation for Science and Technology.

References

  1. 1.
    Acerbi, E., Mingione, G.: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal. 156(2), 121–140 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Acerbi, E., Mingione, G.: Gradient estimates for the \(p(x)\)-Laplacian system. J. Reine Angew. Math. 584, 117–148 (2005)Google Scholar
  3. 3.
    Almeida, A., Hästö, P.: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258(5), 1628–1655 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Almeida, A., Harjulehto, Hästö, P., Lukkari, T.: Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces, Preprint (2013)Google Scholar
  5. 5.
    Almeida, A., Samko, S.: Characterization of Riesz and Bessel potentials on variable Lebesgue spaces. J. Funct. Spaces Appl. 4(2), 113–144 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Almeida, A., Samko, S.: Pointwise inequalities in variable Sobolev spaces and applications. Z. Anal. Anwend. 26(2), 179–193 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Almeida, A., Samko, S.: Embeddings of variable Hajłasz-Sobolev spaces into Hölder spaces of variable order. J. Math. Anal. Appl. 353(2), 489–496 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bergh, J., Löfström, J.: Interpolation Spaces. An introduction. Springer, Berlin (1976)Google Scholar
  9. 9.
    Besov, O.: Interpolation, embedding, and extension of spaces of functions of variable smoothness, (Russian) Tr. Mat. Inst. Steklova 248 (2005), Issled. po Teor. Funkts. i Differ. Uravn., 52–63. [Translation in Proc. Steklov Inst. Math. 248 (2005), no. 1, 47–58.]Google Scholar
  10. 10.
    Chen, Y., Levine, S., Rao, R.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Diening, L., Harjulehto, P., Hästö, P., R\(\mathring{\rm u}\)žička, M.: Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)Google Scholar
  12. 12.
    Diening, L., Hästö, P., Nekvinda, A.: Open problems in variable exponent Lebesgue and Sobolev spaces. In: Drabek and Rákosník (eds.) FSDONA04 Proceedings, Milovy, Czech Republic, pp. 38–58 (2004)Google Scholar
  13. 13.
    Diening, L., Hästö, P., Roudenko, S.: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256(6), 1731–1768 (2009)Google Scholar
  14. 14.
    Fan, X.-L.: Global \(C^{1,\alpha }\) regularity for variable exponent elliptic equations in divergence form. J. Differ. Eqn. 235(2), 397–417 (2007)CrossRefzbMATHGoogle Scholar
  15. 15.
    Fortini, R., Mugnai, D., Pucci, P.: Maximum principles for anisotropic elliptic inequalities. Nonlinear Anal. 70(8), 2917–2929 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)Google Scholar
  17. 17.
    Gurka, P., Harjulehto, P., Nekvinda, A.: Bessel potential spaces with variable exponent. Math. Inequal. Appl. 10(3), 661–676 (2007)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Harjulehto, P., Hästö, P., Latvala, V., Toivanen, O.: Gamma convergence for functionals related to image restoration. Appl. Math. Lett. 26, 56–60 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Harjulehto, P., Hästö, P., Lê, U., Nuortio, M.: Overview of differential equations with non-standard growth. Nonlinear Anal. 72(12), 4551–4574 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, III, IV. Springer, New York (1985)Google Scholar
  21. 21.
    Kempka, H.: 2-microlocal Besov and Triebel-Lizorkin spaces of variable integrability. Rev. Mat. Complut. 22(1), 227–251 (2009)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Kempka, H.: Atomic, molecular and wavelet decomposition of 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability, Funct. Approx. Comment. Math. 43 (2010), part 2, 171–208Google Scholar
  23. 23.
    Kempka, H., Vybíral, J.: A note on the spaces of variable integrability and summability of Almeida and Hästö. Proc. Am. Math. Soc. 141(9), 3207–3212 (2013)CrossRefzbMATHGoogle Scholar
  24. 24.
    Kempka, H., Vybíral, J.: Spaces of variable smoothness and integrability: characterizations by local means and ball means of differences. J. Fourier Anal. Appl. 18(4), 852–891 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Kempka, H., Vybíral, J.: Lorentz spaces with variable exponents. Math. Nachr. (2013, to appear)Google Scholar
  26. 26.
    Kopaliani, T.: Interpolation theorems for variable exponent Lebesgue spaces. J. Funct. Anal. 257(11), 3541–3551 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{1, p(x)}\). Czechoslovak Math. J. 41(116), 592–618 (1991)MathSciNetGoogle Scholar
  28. 28.
    Leopold, H.-G.: On Besov spaces of variable order of differentiation. Z. Anal. Anwendungen 8(1), 69–82 (1989)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Leopold, H.-G.: Interpolation of Besov spaces of variable order of differentiation. Arch. Math. 53, 178–187 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Leopold, H.-G.: On function spaces of variable order of differentiation. Forum Math. 3, 633–644 (1991)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Li, F., Li, Z., Pi, L.: Variable exponent functionals in image restoration. Appl. Math. Comput. 216(3), 870–882 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Orlicz, W.: Über konjugierte Exponentenfolgen. Studia Math. 3, 200–212 (1931)Google Scholar
  33. 33.
    Rabinovich, V., Samko, S.: Boundedness and Fredholmness of pseudodifferential operators in variable exponent spaces. Int. Eqn. Oper. Theory 60(4), 507–537 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    R\(\mathring{\rm u}\)žička, M.: Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, 1748. Springer, Berlin (2000)Google Scholar
  35. 35.
    Schneider, R., Reichmann, O., Schwab, C.: Wavelet solution of variable order pseudodifferential equations. Calcolo 47(2), 65–101 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Taylor, M.E.: Partial Differential Equations II, Applied Mathematical Sciences, vol. 116. Springer, New York (1996)CrossRefGoogle Scholar
  37. 37.
    Taylor, M.E.: Tools for PDE: Pseudodifferential Operators, Paradifferential Operators and Layer Potentials, Mathematical Surveys and Monographs, vol. 81. AMS, Providence (2000)Google Scholar
  38. 38.
    Triebel, H.: Theory of Function Spaces, Monographs in Mathematics 78. Birkhäuser Verlag, Basel (1983)CrossRefGoogle Scholar
  39. 39.
    Triebel, H.: Theory of Function Spaces II, Monographs in Mathematics 84. Birkhäuser Verlag, Basel (1992)CrossRefGoogle Scholar
  40. 40.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)zbMATHGoogle Scholar
  41. 41.
    Vybíral, J.: Sobolev and Jawerth embeddings for spaces with variable smoothness and integrability. Ann. Acad. Sci. Fenn. Math. 34(2), 529–544 (2009)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Xu, J.-S.: Variable Besov and Triebel-Lizorkin spaces. Ann. Acad. Sci. Fenn. Math. 33, 511–522 (2008)zbMATHMathSciNetGoogle Scholar
  43. 43.
    Xu, J.-S.: The relation between variable Bessel potential spaces and Triebel-Lizorkin spaces. Int. Transforms Spec. Funct. 19(8), 599–605 (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2013

Authors and Affiliations

  1. 1.Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Department of Mathematical SciencesUniversity of OuluOulufFinland

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