Boundedness of the Higher Order Commutators of Marcinkiewicz Integral Operator on Grand Variable Herz-Morrey Spaces
Main Article Content
Abstract
In this paper, the authors prove the boundedness of higher order commutators of Marcinkiewicz integral operator under some proper assumptions on grand variable Herz-Morrey spaces. Then we obtain the estimates for the Marcinkiewicz fractional operator of variable order in grand variable Herz spaces.
Article Details
References
- D.V. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces, Springer, (2013).
- L. Diening, P. H{"a}st{"o}, A. Nekvinda, Open Problems in Variable Exponent Lebesgue and Sobolev Spaces, in: FSDONA04 Proceedings, pp. 38–52, (2004).
- L. Diening, P. Harjulehto, P. Hästö, M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, (2011).
- M. Izuki, Boundedness of Vector-Valued Sublinear Operators on Herz-Morrey Spaces With Variable Exponent, Math. Sci. Res. J. 13 (2009), 243–253.
- M. Izuki, Boundedness of Commutators on Herz Spaces with Variable Exponent, Rend. Circ. Mat. Palermo 59 (2010), 199–213. https://doi.org/10.1007/s12215-010-0015-1.
- V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces, Vol. 1: Variable Exponent Lebesgue and Amalgam Spaces, Vol. 248, Springer, Cham, 2016. https://doi.org/10.1007/978-3-319-21015-5.
- V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces, Vol. 2: Variable Exponent Hölder, Morrey–campanato and Grand Spaces, Vol. 249, Springer, Cham, 2016. https://doi.org/10.1007/978-3-319-21018-6.
- V. Kokilashvili, S. Samko, On Sobolev Theorem for Riesz-Type Potentials in Lebesgue Spaces with Variable Exponent, Z. Anal. Anwend. 22 (2003), 899–910. https://doi.org/10.4171/zaa/1178.
- A. Meskhi, H. Rafeiro, M.A. Zaighum, On the Boundedness of Marcinkiewicz Integrals on Continual Variable Exponent Herz Spaces, Georgian Math. J. 26 (2017), 105–116. https://doi.org/10.1515/gmj-2017-0050.
- M. Sultan, B. Sultan, A Note on the Boundedness of Marcinkiewicz Integral Operator on Continual Herz-Morrey Spaces, Filomat 39 (2025), 2017–2027.
- H. Rafeiro, S. Samko, Riesz Potential Operator in Continual Variable Exponents Herz Spaces, Math. Nachr. 288 (2014), 465–475. https://doi.org/10.1002/mana.201300270.
- M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin, 2000. https://doi.org/10.1007/bfb0104029.
- S. Samko, Variable Exponent Herz Spaces, Mediterr. J. Math. 10 (2013), 2007–2025. https://doi.org/10.1007/s00009-013-0285-x.
- M. Sultan, B. Sultan, A. Khan, T. Abdeljawad, Boundedness of Marcinkiewicz Integral Operator of Variable Order in Grand Herz-Morrey Spaces, AIMS Math. 8 (2023), 22338–22353. https://doi.org/10.3934/math.20231139.
- B. Sultan, M. Sultan, Boundedness of Higher Order Commutators of Hardy Operators on Grand Herz-Morrey Spaces, Bull. Des Sci. Mathématiques 190 (2024), 103373. https://doi.org/10.1016/j.bulsci.2023.103373.
- M. Sultan, B. Sultan, A. Hussain, Grand Herz–morrey Spaces with Variable Exponent, Math. Notes 114 (2023), 957–977. https://doi.org/10.1134/s0001434623110305.
- M. Sultan, B. Sultan, A. Aloqaily, N. Mlaiki, Boundedness of Some Operators on Grand Herz Spaces with Variable Exponent, AIMS Math. 8 (2023), 12964–12985. https://doi.org/10.3934/math.2023653.
- B. Sultan, F. Azmi, M. Sultan, M. Mehmood, N. Mlaiki, Boundedness of Riesz Potential Operator on Grand Herz-Morrey Spaces, Axioms 11 (2022), 583. https://doi.org/10.3390/axioms11110583.
- B. Sultan, M. Sultan, Boundedness of Commutators of Rough Hardy Operators on Grand Variable Herz Spaces, Forum Math. 36 (2023), 717–733. https://doi.org/10.1515/forum-2023-0152.
- A. Hussain, G. Gao, Multilinear Singular Integrals and Commutators on Herz Space with Variable Exponent, ISRN Math. Anal. 2014 (2014), 626327. https://doi.org/10.1155/2014/626327.
- A. Hussain, I. Khan, A. Mohamed, Variable Herz–morrey Estimates for Rough Fractional Hausdorff Operator, J. Inequal. Appl. 2024 (2024), 33. https://doi.org/10.1186/s13660-024-03110-8.
- A. Ajaib, A. Hussain, Weighted Cbmo Estimates for Commutators of Matrix Hausdorff Operator on the Heisenberg Group, Open Math. 18 (2020), 496–511. https://doi.org/10.1515/math-2020-0175.
- A. Hussain, A. Ajaib, Some Results for the Commutators of Generalized Hausdorff Operator, J. Math. Inequal. (2019), 1129–1146. https://doi.org/10.7153/jmi-2019-13-80.
- J. Younas, A. Hussain, H. Alhazmi, A.F. Aljohani, I. Khan, Bmo Estimates for Commutators of the Rough Fractional Hausdorff Operator on Grand-Variable-Herz-Morrey Spaces, AIMS Math. 9 (2024), 23434–23448. https://doi.org/10.3934/math.20241139.
- B. Sultan, M. Sultan, I. Khan, On Sobolev Theorem for Higher Commutators of Fractional Integrals in Grand Variable Herz Spaces, Commun. Nonlinear Sci. Numer. Simul. 126 (2023), 107464. https://doi.org/10.1016/j.cnsns.2023.107464.
- B. Sultan, M. Sultan, Sobolev-type Theorem for Commutators of Hardy Operators in Grand Herz Spaces, Ukr. Mat. Zhurnal 76 (2024), 1052–1068. https://doi.org/10.3842/umzh.v76i7.7546.
- M. Sultan, B. Sultan, R.E. Castillo, Weighted Composition Operator on Gamma Spaces with Variable Exponent, J. Pseudo-Differ. Oper. Appl. 15 (2024), 46. https://doi.org/10.1007/s11868-024-00619-w.
- M. Sultan, B. Sultan, A Note on the Boundedness of Higher Order Commutators on Fractional Integrals in Grand Variable Herz-Morrey Spaces, Kragujev. J. Math. 50 (2026), 1063–1080.
- B. Sultan, M. Sultan, A. Khan, T. Abdeljawad, Boundedness of Commutators of Variable Marcinkiewicz Fractional Integral Operator in Grand Variable Herz Spaces, J. Inequal. Appl. 2024 (2024), 93. https://doi.org/10.1186/s13660-024-03169-3.
- B. Sultan, M. Sultan, Q. Zhang, N. Mlaiki, Boundedness of Hardy Operators on Grand Variable Weighted Herz Spaces, AIMS Math. 8 (2023), 24515–24527. https://doi.org/10.3934/math.20231250.
- B. Sultan, M. Sultan, A. Hussain, Boundedness of the Bochner–riesz Operators on the Weighted Herz-Morrey Type Hardy Spaces, Complex Anal. Oper. Theory 19 (2025), 49. https://doi.org/10.1007/s11785-025-01671-0.
- B. Sultan, A. Hussain, M. Sultan, Chracterization of Generalized Campanato Spaces with Variable Exponents via Fractional Integrals, J. Pseudo-Differ. Oper. Appl. 16 (2025), 22. https://doi.org/10.1007/s11868-024-00651-w.
- B. Sultan, M. Sultan, M. Mehmood, F. Azmi, M.A. Alghafli, N. Mlaiki, Boundedness of Fractional Integrals on Grand Weighted Herz Spaces with Variable Exponent, AIMS Math. 8 (2023), 752–764. https://doi.org/10.3934/math.2023036.
- B. Sultan, F.M. Azmi, M. Sultan, T. Mahmood, N. Mlaiki, N. Souayah, Boundedness of Fractional Integrals on Grand Weighted Herz–morrey Spaces with Variable Exponent, Fractal Fract. 6 (2022), 660. https://doi.org/10.3390/fractalfract6110660.
- H. Wang, Commutators of Marcinkiewicz Integrals on Herz Spaces with Variable Exponent, Czechoslov. Math. J. 66 (2016), 251–269. https://doi.org/10.1007/s10587-016-0254-1.
- J. Wu, W. Zhao, Boundedness for Fractional Hardy-Type Operator on Variable-Exponent Herz–morrey Spaces, Kyoto J. Math. 56 (2016), 831–845. https://doi.org/10.1215/21562261-3664932.