Skip to main content
SpringerLink
Log in

Maximal and Calderón–Zygmund operators in grand variable Lebesgue spaces

  • Original Paper
  • Published:
Banach Journal of Mathematical Analysis Aims and scope Submit manuscript

Abstract

New Banach function spaces \(L^{p(\cdot ),\theta }(X)\) unifying grand Lebesgue spaces and variable Lebesgue spaces are introduced by Kokilashvili and Meskhi. The spaces and operators are defined on quasi-metric finite measure spaces with doubling condition (spaces of homogeneous type). Weighted inequalities with power-type weights in \(L^{p(\cdot ),\theta }(X)\) are obtained for Hardy–Littlewood maximal operators, singular integral operators, and commutators of singular integrals. To obtain these results, a weighted extrapolation theorem on \(L^{p(\cdot ),\theta }(X)\) is established, which has an independent interest.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bramanti, M., Cerutti, M.C.: Commutators of singular integrals on homogeneous spaces. Boll. Unione Mat. Ital. VII. Ser. B 4, 843–884 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Coifman, R.R., Weiss, G.: Analyse Harmonique Non-commutative Sur Certains Espaces Homogènes, Lecture Notes in Mathematics, vol. 242. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  4. Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis. Birkhäuser, New York (2013)

    Book  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Cruz-Uribe, D., Cummings, J.: Weighted norm inequalities for the maximal operator on \(L^{p(\cdot )}\) over spaces of homogeneous type. Ann. Fenn. Math. 47, 457–488 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cruz-Uribe, D., Shukla, P.: The boundedness of fractional maximal operators on variable Lebesgue spaces over spaces of homogeneous type. Stud. Math. 242, 109–139 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cruz-Uribe, D., Wang, L.A.: Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. Trans. Am. Math. Soc. 369, 1205–1235 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Diening, L., Karlovych, O., Shargorodsky, E.: On interpolation of reflexive variable Lebesgue spaces on which the Hardy-Littlewood maximal operator is bounded. Georgian Math. J. 29, 347–352 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. Diening, L., Hastö, P.: Muckenhoupt weights in variable exponent spaces (2008) (preprint)

  13. Duoandikoetxea, J.: Fourier Analysis, Graduate Studies in Mathematics, vol. 29. American Mathematical Society (AMS), Providence (2001)

    Google Scholar 

  14. Dong, B., Xu, J.: Herz-Morrey type Besov and Triebel-Lizorkin spaces with variable exponents. Banach J. Math. Anal. 9, 75–101 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Edmunds, D., Kokilashvili, V., Meskhi, A.: Embeddings in grand variable exponent function spaces. Results Math. 76, 1–27 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fiorenza, A., Gupta, B., Jain, P.: The maximal theorem for weighted grand Lebesgue spaces. Stud. Math. 188, 123–133 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Greco, L., Iwaniec, T., Sbordone, C.: Inverting the p-harmonic operator. Manuscr. Math. 92, 249–258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Harjulehto, P., Hästö, P., Latvala, V., Toivanen, O.: Critical variable exponent functionals in image restoration. Appl. Math. Lett. 26, 56–60 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Harjulehto, P., Hästö, P., Pere, M.: Variable exponent Lebesgue spaces on metric spaces: the Hardy-Littlewood maximal operator. Real Anal. Exch. 30, 87–103 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Han, Y., Müller, D., Yang, D.: A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr. Appl. Anal. (2008)

  21. Hytönen, T., Pérez, C., Rela, E.: Sharp reverse Hölder property for \(A_{\infty }\) weights on spaces of homogeneous type. J. Funct. Anal. 263, 3883–3899 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Ration. Mech. Anal. 119, 129–143 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kokilashvili, V., Meskhi, A.: Maximal and Calderón-Zygmund operators in grand variable exponent Lebesgue spaces. Georgian Math. J. 21, 447–461 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kokilashvili, V., Meskhi, A.: Maximal and singular integral operators in weighted grand variable exponent Lebesgue spaces. Ann. Funct. Anal. 12, 1–29 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kokilashvili, V., Meskhi, A., Rafeiro, H., Samko, S.: Integral Operators in Non-Standard Function Spaces: Variable Exponent Lebesgue and Amalgam Spaces, Operator Theory: Advances and Applications. Birkhäuser, Basel (2016)

    MATH  Google Scholar 

  26. Kokilashvili, V., Samko, S.: Maximal and fractional operators in weighted \(L^{p(x)}\) spaces. Rev. Mat. Iberoam. 20, 493–515 (2004)

    Article  MATH  Google Scholar 

  27. Kokilashvili, V., Samko, S.: The maximal operator in weighted variable exponent spaces on metric spaces. Georgian Math. J. 15, 683–712 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Macías, R., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pradolini, G., Salinas, O.: Commutators of singular integrals on spaces of homogeneous type. Czech. Math. J. 57, 75–93 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Samko, S., Umarkhadzhiev, S.: On Iwaniec-Sbordone spaces on sets which may have infinite measure. Azerb. J. Math. 1, 67–84 (2011)

    MathSciNet  MATH  Google Scholar 

  31. Wang, H., Xu, J., Tan, J.: Boundedness of multilinear singular integrals on central Morrey spaces with variable exponents. Front. Math. China 15, 1011–1034 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yan, X., Yang, D., Yuan, W., Zhuo, C.: Variable weak Hardy spaces and their applications. J. Funct. Anal. 271, 2822–2887 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang, D., Zhou, Y.: New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. Manuscr. Math. 134, 59–90 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yang, D., Zhuo, C., Yuan, W.: Triebel-Lizorkin type spaces with variable exponents. Banach J. Math. Anal. 9, 146–202 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhuo, C., Sawano, Y., Yang, D.: Hardy spaces with variable exponents on RD-spaces and applications. Diss. Math. (Rozprawy Mat.) 520, 74 (2016)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the referees for their careful reading of the paper and for their useful comments and suggestions. This project is supported by the National Natural Science Foundation of China (Nos. 12261083, 12161083), the Natural Science Foundation of Xinjiang Uyghur Autonomous Region (No. 2020D01C048), and Xinjiang Key Laboratory of Applied Mathematics (No. XJDX1401).

Author information

Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830017, People’s Republic of China

    Shuai Yang, Jiawei Sun & Baode Li

Authors
  1. Shuai Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiawei Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Baode Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Baode Li.

Additional information

Communicated by Dachun Yang.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, S., Sun, J. & Li, B. Maximal and Calderón–Zygmund operators in grand variable Lebesgue spaces. Banach J. Math. Anal. 17, 46 (2023). https://doi.org/10.1007/s43037-023-00272-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43037-023-00272-3

Keywords

Mathematics Subject Classification

Search

Navigation