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.
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References
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)
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)
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)
Bastero, J., Milman, M., Ruiz, F.J.: Commutators for the maximal and sharp functions. Proc. Am. Math. Soc. 128, 3329–3334 (2000)
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)
Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer Monographs in Mathematics, Springer, Berlin (2007)
Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103(3), 611–635 (1976)
Cruz-Uribe, D., Fiorenza, A.: Endpoint estimate and weighted norm inequalities for commutators of fractional integrals. Publ. Mat. 47, 103–131 (2003)
Deringoz, F., Guliyev, V.S., Hasanov, S.G.: Commutators of fractional maximal operator on generalized Orlicz–Morrey spaces. Positivity 22(1), 141–158 (2018)
Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups, Mathematical Notes 28. Princeton Univ. Press, Princeton (1982)
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)
Grafakos, L.: Modern Fourier Analysis, 2nd edn. Springer, New York, NY (2009)
Guliyev, V.S., Deringoz, F., Hasanov, S.G.: Riesz potential and its commutators on Orlicz spaces. J. Inequal. Appl. 1, 1–18 (2017)
Guliyev, V.S., Deringoz, F., Hasanov, S.G.: Commutators of fractional maximal operator on Orlicz spaces. Math. Notes 104(4), 498–507 (2018)
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)
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)
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)
Ho, K.P.: Characterization of \(BMO\) in terms of rearrangement-invariant Banach function spaces. Expo. Math. 27, 363–372 (2009)
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)
Janson, S.: Mean oscillation and commutators of singular integral operators. Ark. Mat. 16, 263–270 (1978)
John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)
Kokilashvili, V.M., Krbec, M.M.: Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific, Singapore (1991)
Long, R., Yang, L.: \(BMO\) functions in spaces of homogeneous type. Sci. Sin. Ser. A 27(7), 695–708 (1984)
Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. M. Dekker Inc, New York (1991)
Segovia, C., Torrea, J.L.: Weighted inequalities for commutators of fractional and singular integrals. Publ. Mat. 35, 209–235 (1991)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton Univ. Press, Princeton (1993)
Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100. Cambridge University, Cambridge (1992)
Zhang, P., Wu, J.: Commutators of the fractional maximal functions. Acta Math. Sin. (Chin. Ser.) 52, 1235–1238 (2009)
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)
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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.
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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
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DOI: https://doi.org/10.1007/s00025-021-01578-0