Skip to main content

Wavelet Characterization of Local Muckenhoupt Weighted Sobolev Spaces with Variable Exponents

Abstract

The goal of this paper is to define local weighted variable Sobolev spaces of fractional and negative order and their characterization by wavelets. We first consider local weighted variable Sobolev spaces by means of weak derivatives and obtain a wavelet characterization for these spaces. Using the Bessel potentials, we next define local weighted variable Sobolev spaces of fractional order. We show that Sobolev spaces obtained by weak derivatives and those by the Bessel potentials coincide. Finally, using duality, we define local weighted variable Sobolev spaces with negative order. We also show that local weighted variable Sobolev spaces are closed under complex interpolation. Some examples are given including the applications to weighted uniformly local Lebesgue spaces with variable exponents and periodic function spaces as a by-product, although the exponent is constant.

This is a preview of subscription content, access via your institution.

References

  1. Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer, Berlin, New York (1976)

  2. Bertrandias, J., Datry, C., Dupuis, C.: Unions et intersections d’espaces \(L^p\) invariantes par translation ou convolution. Ann. Inst. Fourier (Grenoble) 28(2), 53–84 (1978)

  3. Calderón, A.P.: Intermediate spaces and interpolation, the complex method, Studia Math. 24, 113–190. no. 1, 46–56 (1964)

  4. Cowling, M., Meda, S., Pasquale, R.: Riesz potentials and amalgams. Ann. Inst. Fourier (Grenoble) 49(4), 1345–1367 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  5. Cruz-Uribe, D.V., Diening, L., Hästö, P.: The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal. 14, 361–374 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  6. Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces, Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg (2013)

    MATH  Book  Google Scholar 

  7. Cruz-Uribe, D.V., Fiorenza, A., Martell, J.M., Pérez, C.: The boundedness of classical operators on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Cruz-Uribe, D.V., Fiorenza, A., Neugebauer, C.J.: The maximal function on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 28, 223–238 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Cruz-Uribe, D.V., Fiorenza, A., Neugebauer, C.J.: Corrections to: “The maximal function on variable \(L^p\) spaces” [Ann. Acad. Sci. Fenn. Math. 28 (1) (2003), 223–238.], Ann. Acad. Sci. Fenn. Math. 29, 247–249 (2004)

  10. Cruz-Uribe, D.V., Fiorenza, A., Neugebauer, C.J.: Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. J. Math. Anal. Appl. 394, 744–760 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  11. Cruz-Uribe, D.V., Fiorenza, A., Martell, J.M., Pérez, C.: Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215. Birkhäuser/Springer, Basel (2011)

    MATH  Google Scholar 

  12. Daubechies, I.: Orthonormal basis of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988)

    MATH  Article  Google Scholar 

  13. Diening, L.: Maximal function on generalized Lebesgue spaces \(L^{p(\cdot )}\). Math. Inequal. Appl. 7, 245–253 (2004)

    MathSciNet  MATH  Google Scholar 

  14. Diening, L., Hästö, P., Roudenko, S.: Spaces of variable integrability and differentiability. J. Funct. Anal. 256, 1731–1768 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  15. Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)

  16. Duoandikoetxea, J.: Fourier Analysis. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence (2001)

  17. Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1), 34–170 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  18. Fournier, J., Stewart, J.: Amalgams of \(L^p\) and \(l^q\). Bull . Am. Math. Soc. (N.S.) 13(1), 1–21 (1985)

    MATH  Article  MathSciNet  Google Scholar 

  19. Garrigós, G., Seeger, A., Ullrich, T.: The Haar System as a Schauder Basis in Spaces of Hardy–Sobolev Type. J. Four. Anal Appl. 24, 1319–1339 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  20. Grafakos, L.: Modern Fourier Analysis, Graduate texts in Mathematics, vol. 250. Springer, New York (2014)

    Google Scholar 

  21. Hästö, P.: Local-to-global results in variable exponent spaces. Math. Res. Lett. 16(2), 263–278 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  22. Hernández, E., Weiss, G.: A First Course on Wavelets. CRC Press, Boca Raton (1996)

    MATH  Book  Google Scholar 

  23. Holland, F.: Harmonic analysis on amalgams of \(L^p\) and \(\ell ^q\). J. Lond. Math. Soc. (2) 10, 295–305 (1975)

    MATH  Article  MathSciNet  Google Scholar 

  24. Izuki, M.: Wavelets and modular inequalities in variable \(L^p\) spaces. Georgian Math. J. 15, 281–293 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  25. Izuki, M.: The characterizations of weighted Sobolev spaces by wavelets and scaling functions. Taiwan. J. Math. 13, 467–492 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  26. Izuki, M., Nakai, E., Sawano, Y.: Function spaces with variable exponents—an introduction. Sci. Math. Jpn. 77, 187–315 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Izuki, M., Nakai, E., Sawano, Y.: Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponents. Ann. Acad. Sci. Fenn. Math. 40, 551–571 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  28. Izuki, M., Nogayama, T., Noi, T., Sawano, Y.: Wavelet characterization of local Muckenhoupt weighted Lebesgue spaces with variable exponents. Nonlinear Anal. 198, 111930 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  29. Izuki, M., Sawano, Y.: The Haar wavelet characterization of weighted Herz spaces and greediness of the Haar wavelet basis. J. Math. Anal. Appl. 362(1), 140–155 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  30. Kikuchi, N., Nakai, E., Tomita, N., Yabuta, K., Yoneda, T.: Calderón–Zygmund operators on amalgam spaces and in the discrete case. J. Math. Anal. Appl. 335, 198–212 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  31. Kopaliani, T.S.: Greediness of the wavelet system in \(L^{p(t)}({\mathbb{R}})\) spaces. East J. Approx. 14, 59–67 (2008)

    MathSciNet  MATH  Google Scholar 

  32. Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslovak Math. J. 41, 592–618 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  33. Lemarié-Rieusset, P.G.: Ondelettes et poids de Muckenhoupt. Studia Math. 108, 127–147 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  34. Lemarié-Rieusset, P.G., Malgouyres, G.: Support des fonctions de base dans une analyse multi-résolution. C. R. Acad. Sci. Paris 313, 377–380 (1991)

    MathSciNet  MATH  Google Scholar 

  35. Małecka, A.: Haar functions in weighted Besov and Triebel–Lizorkin spaces. J. Approx. Theory 200, 1–27 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  36. Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  37. Nakai, E., Tomita, N., Yabuta, K.: Density of the set of all infinitely differentiable functions with compact support in weighted Sobolev spaces. Sci. Math. Jpn. 60, 121–127 (2004)

    MathSciNet  MATH  Google Scholar 

  38. Nakano, H.: Modulared Semi-Ordered Linear Spaces. Maruzen Co., Ltd, Tokyo (1950)

    MATH  Google Scholar 

  39. Nakano, H.: Topology of Linear Topological Spaces. Maruzen Co., Ltd, Tokyo (1951)

    Google Scholar 

  40. Nogayama, T., Sawano, Y.: Local Muckenhoupt class for variable exponents. J. Inequal. Appl. (2021). https://doi.org/10.1186/s13660-021-02601-2

    MathSciNet  Article  MATH  Google Scholar 

  41. Orlicz, W.: Über konjugierte Exponentenfolgen. Studia Math. 3, 200–212 (1931)

    MATH  Article  Google Scholar 

  42. Rychkov, V.S.: Littlewood–Paley theory and function spaces with \(A^{\rm loc}_p\) weights. Math. Nachr. 224, 145–180 (2001)

    MathSciNet  Article  Google Scholar 

  43. Sawano, Y.: Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integr. Equ. Oper. Theory 77(1), 123–148 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  44. Sawano, Y.: Theory of Besov Spaces, Development in Mathematics, vol. 56. Springer, Singapore (2018)

    MATH  Book  Google Scholar 

  45. Shestakov, V.A.: On complex interpolation of Banach spaces of measurable functions. Vestnik Leningrad. Univ. 19, 64–68 (1974)

    MathSciNet  MATH  Google Scholar 

  46. Sharapudinov, I.I.: Approximation of functions in \(L^{p(x)}_{2\pi }\) by trigonometric polynomials. Izv. RAN. Ser. Mat. 77(2), 197–224 (2013)

    Article  Google Scholar 

  47. Sickel, W., Skrzypczak, L., Vybíral, J.: Complex interpolation of weighted Besov and Lizorkin–Triebel spaces. Acta. Math. Sin.-English Ser 30, 1297–1323 (2014). https://doi.org/10.1007/s10114-014-2762-y

    MathSciNet  Article  MATH  Google Scholar 

  48. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  49. Triebel, H.: Theory of Function Spaces II. Birkhäuser, New York (1992)

    MATH  Book  Google Scholar 

  50. Wojtaszczyk, P.: A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge (1997)

    MATH  Book  Google Scholar 

  51. Wojciechowska, A.: Multidimensional wavelet bases in Besov and Triebel–Lizorkin spaces, Ph.D. thesis. Adam Mickiewicz University Poznań, Poznań (2012)

  52. Zhu, K.: Operator Theory in Function Spaces, vol. 138. American Mathematical Society, Providence (2007)

    MATH  Book  Google Scholar 

Download references

Acknowledgements

First of all, the authors thank the anonymous referees for his/her comments on this paper, which improved readability. Mitsuo Izuki was partially supported by Grand-in-Aid for Scientific Research (C) (Grant No. 15K04928), for Japan Society for the Promotion of Science. Toru Nogayama was partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists (Grant No. 20J10403). Takahiro Noi was partially supported by Grand-in-Aid for Young Scientists (B) (Grant No. 17K14207) for Japan Society for the Promotion of Science. Yoshihiro Sawano was partially supported by Grand-in-Aid for Scientific Research (C) (Grant No.  19K03546), for Japan Society for the Promotion of Science. This work was partly supported by Osaka City University Advanced Mathematical Institute: MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Toru Nogayama.

Additional information

Communicated by Wolfgang Dahmen.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Izuki, M., Nogayama, T., Noi, T. et al. Wavelet Characterization of Local Muckenhoupt Weighted Sobolev Spaces with Variable Exponents. Constr Approx (2022). https://doi.org/10.1007/s00365-022-09573-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00365-022-09573-6

Keywords

  • Variable exponent
  • Wavelet
  • Sobolev spaces
  • Local Muckenhoupt weight

Mathematics Subject Classification

  • 42B35
  • 42C40