Skip to main content

Some Characterizations of BMO Spaces via Commutators in Orlicz Spaces on Stratified Lie Groups

Abstract

In the paper we study the fractional maximal commutators \(M_{b,\alpha }\) and the commutators of the fractional maximal operator \([b, M_{\alpha }]\) in the Orlicz spaces \(L^{\Phi }(\mathbb {G})\) on any stratified Lie group \(\mathbb {G}\). We give necessary and sufficient conditions for the boundedness of the operators \(M_{b,\alpha }\) and \([b, M_{\alpha }]\) on Orlicz spaces \(L^{\Phi }(\mathbb {G})\) when b belongs to \(BMO(\mathbb {G})\) spaces, whereby some new characterizations for certain subclasses of \(BMO(\mathbb {G})\) spaces are obtained.

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

References

  1. Agcayazi, M., Gogatishvili, A., Koca, K., Mustafayev, R.: A note on maximal commutators and commutators of maximal functions. J. Math. Soc. Jpn. 67(2), 581–593 (2015)

    MathSciNet  Article  Google Scholar 

  2. Akbulut, A., Burenkov, V.I., Guliyev, V.S.: Anisotropic fractional maximal commutators with BMO functions on anisotropic Morrey-type spaces. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. 40(4), 13–32 (2020)

    MathSciNet  Google Scholar 

  3. Aykol, C., Armutcu, H., Omarova, M.N.: Maximal commutator and commutator of maximal function on modified Morrey spaces. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. 36(1), 29–35 (2016)

    MathSciNet  Google Scholar 

  4. Bastero, J., Milman, M., Ruiz, F.J.: Commutators for the maximal and sharp functions. Proc. Am. Math. Soc. 128, 3329–3334 (2000)

    MathSciNet  Article  Google Scholar 

  5. Bonami, A., Iwaniec, T., Jones, P., Zinsmeister, M.: On the product of functions in \(BMO\) and \(H_1\). Ann. Inst. Fourier Grenoble 57(5), 1405–1439 (2007)

    MathSciNet  Article  Google Scholar 

  6. Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer Monographs in Mathematics, Springer, Berlin (2007)

    MATH  Google Scholar 

  7. Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103(3), 611–635 (1976)

    MathSciNet  Article  Google Scholar 

  8. Cruz-Uribe, D., Fiorenza, A.: Endpoint estimate and weighted norm inequalities for commutators of fractional integrals. Publ. Mat. 47, 103–131 (2003)

    MathSciNet  Article  Google Scholar 

  9. Deringoz, F., Guliyev, V.S., Hasanov, S.G.: Commutators of fractional maximal operator on generalized Orlicz–Morrey spaces. Positivity 22(1), 141–158 (2018)

    MathSciNet  Article  Google Scholar 

  10. Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups, Mathematical Notes 28. Princeton Univ. Press, Princeton (1982)

    MATH  Google Scholar 

  11. Garcia-Cuerva, J., Harboure, E., Segovia, C., Torrea, J.L.: Weighted norm inequalities for commutators of strongly singular integrals. Indiana Univ. Math. J. 40, 1397–1420 (1991)

    MathSciNet  Article  Google Scholar 

  12. Grafakos, L.: Modern Fourier Analysis, 2nd edn. Springer, New York, NY (2009)

    Book  Google Scholar 

  13. Guliyev, V.S., Deringoz, F., Hasanov, S.G.: Riesz potential and its commutators on Orlicz spaces. J. Inequal. Appl. 1, 1–18 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Guliyev, V.S., Deringoz, F., Hasanov, S.G.: Commutators of fractional maximal operator on Orlicz spaces. Math. Notes 104(4), 498–507 (2018)

    MathSciNet  Article  Google Scholar 

  15. Guliyev, V.S., Deringoz, F., Hasanov, S.G.: Fractional maximal function and its commutators on Orlicz spaces. Anal. Math. Phys. 9(1), 165–179 (2019)

    MathSciNet  Article  Google Scholar 

  16. Guliyev, V.S., Ekincioglu, I., Kaya, E., Safarov, Z.: Characterizations for the fractional maximal operator and its commutators in generalized Morrey spaces on Carnot groups. Integral Transforms Spec. Funct. 30(6), 453–470 (2019)

    MathSciNet  Article  Google Scholar 

  17. Guliyev, V.S.: Commutators of the fractional maximal function in generalized Morrey spaces on Carnot groups. Complex Var. Elliptic Equ. 66(6–7), 893–909 (2021)

    MathSciNet  Article  Google Scholar 

  18. Ho, K.P.: Characterization of \(BMO\) in terms of rearrangement-invariant Banach function spaces. Expo. Math. 27, 363–372 (2009)

    MathSciNet  Article  Google Scholar 

  19. Hu, G., Yang, D.: Maximal commutators of \(BMO\) functions and singular integral operators with non-smooth kernels on spaces of homogeneous type. J. Math. Anal. Appl. 354, 249–262 (2009)

    MathSciNet  Article  Google Scholar 

  20. Janson, S.: Mean oscillation and commutators of singular integral operators. Ark. Mat. 16, 263–270 (1978)

    MathSciNet  Article  Google Scholar 

  21. John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)

    MathSciNet  Article  Google Scholar 

  22. Kokilashvili, V.M., Krbec, M.M.: Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific, Singapore (1991)

    Book  Google Scholar 

  23. Long, R., Yang, L.: \(BMO\) functions in spaces of homogeneous type. Sci. Sin. Ser. A 27(7), 695–708 (1984)

    MathSciNet  MATH  Google Scholar 

  24. Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. M. Dekker Inc, New York (1991)

    MATH  Google Scholar 

  25. Segovia, C., Torrea, J.L.: Weighted inequalities for commutators of fractional and singular integrals. Publ. Mat. 35, 209–235 (1991)

    MathSciNet  Article  Google Scholar 

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

    MATH  Google Scholar 

  27. Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100. Cambridge University, Cambridge (1992)

    MATH  Google Scholar 

  28. Zhang, P., Wu, J.: Commutators of the fractional maximal functions. Acta Math. Sin. (Chin. Ser.) 52, 1235–1238 (2009)

    MathSciNet  MATH  Google Scholar 

  29. Zhang, P., Wu, J., Sun, J.: Commutators of some maximal functions with Lipschitz function on Orlicz spaces. Mediterr. J. Math. 15(6), 1–13 (2018)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The author thanks the referee(s) for careful reading the paper and useful comments. The research of author was partially supported by grant of Cooperation Program 2532 TUBITAK—RFBR (RUSSIAN foundation for basic research) (Agreement Number No. 119N455), by Grant of 1st Azerbaijan-Russia Joint Grant Competition (Agreement Number No. EIF-BGM-4-RFTF-1/2017-21/01/1-M-08) and by the RUDN University Strategic Academic Leadership Program.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. S. Guliyev.

Additional information

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

Guliyev, V.S. Some Characterizations of BMO Spaces via Commutators in Orlicz Spaces on Stratified Lie Groups. Results Math 77, 42 (2022). https://doi.org/10.1007/s00025-021-01578-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-021-01578-0

Keywords

  • stratified group
  • Orlicz space
  • fractional maximal function
  • commutator
  • BMO

Mathematics Subject Classification

  • 42B25
  • 46E30