BMO estimate for the higher order commutators of Marcinkiewicz integral operator on grand variable Herz-Morrey spaces

Year 2023, Volume: 72 Issue: 4, 1000 - 1018, 29.12.2023

Babar Sultan , Mehvish Sultan , Ferit Gürbüz

https://doi.org/10.31801/cfsuasmas.1328691

Cited By: 3

Abstract

Let $\mathbb{S}^{n-1}$ denote the unit sphere in $\mathbb{R}^n$ with the normalized Lebesgue measure. Let $\Phi\in L^{r}(\mathbb{S}^{n-1})$ is a homogeneous function of degree zero and $b$ is
a locally integrable function on $\mathbb{R}^n$. In this paper we define the higher order commutators of Marcinkiewicz integral $[b,\mu_{\Phi}]^m$ and prove the boundedness of $[b,\mu_{\Phi}]^m$ under some proper assumptions on grand variable Herz-Morrey spaces $M\dot{K}^{\alpha(.),\beta}_{u,v(.)}(\mathbb{R}^n)$.

Keywords

BMO Spaces, Marcinkiewicz integral operator, grand Herz-Morrey spaces, grand Herz spaces

Supporting Institution

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Project Number

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Thanks

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References

• Asim, M., Hussain, A., Weighted variable Morrey-Herz estimates for fractional Hardy operators, J. Inequal. Appl., 2(2022) (2022) 12pp. https://doi.org/10.1186/s13660-021-02739-z
• Bashir, S., Sultan, B., Hussain, A., Khan, A., Abdeljawad, T., A note on the boundedness of Hardy operators in grand Herz spaces with variable exponent, AIMS Mathematics, 8(9) (2023), 22178–22191. https://doi.org/10.3934/math.20231130
• Diening, L., Harjuletho, P., Hastö, P., Ruzicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, Springer, 2011. https://doi.org/10.1007/978-3-642-18363-8
• Hussain, A., Asim, M., Aslam, M., Jarad, F., Commutators of the fractional Hardy operator on weighted variable Herz-Morrey spaces, J. Funct. Spaces., ID 9705250 (2021), 10 pages. https://doi.org/10.1155/2021/9705250
• Hussain, A., Asim, M., Jarad, F., Variable $\lambda$-Central Morrey space estimates for the fractional Hardy operators and commutators, Journal of Mathematics, ID 5855068 (2022), 12 pp. https://doi.org/10.1155/2022/5855068
• Izuki, M., Boundedness of vector-valued sublinear operators on Herz-Morrey spaces with variable exponents, Math. Sci. J., 13(10) (2009), 243–253.
• Izuki, M., Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math., 36(1) (2010), 33–50.
• Izuki, M., Boundedness of commutators on Herz spaces with variable exponent, Rendiconti del Circolo Matematico di Palermo., 59 (2010), 199–213. https://doi.org/10.1007/s12215-010-0015-1
• Kováčik, O., Rákosník, J., On spaces $L^{p(x)$ and $W^{k,p(x)}$, Czechoslov. Math. J., 41(4) (1991), 592–618.
• Meskhi, A., Maximal functions, potentials and singular integrals in grand Morrey spaces, Complex Variables and Elliptic Equations, 56(10-11) (2011), 1003-1019. https://doi.org/10.1080/17476933.2010.534793
• Muckenhoupt, B., Wheeden, R. L., Weighted norm inequalities for singular and fractional integrals, Trans. Am. Maths Soc., 161 (1971), 249–258.
• Nafis, H., Rafeiro, H., Zaighum, M. A., A note on the boundedness of sublinear operators on grand variable Herz spaces, J. Inequal Appl., 2020(1) (2020), 1–13. https://doi.org/10.1186/s13660-019-2265-6
• Nafis, H., Rafeiro, H., Zaighum, M. A., Boundedness of the Marcinkiewicz integral on grand Herz spaces, J. Math. Ineq., 15(2) (2021), 739–753. https://doi.org/10.7153/jmi-2021-15-52
• Samko, S., Variable exponent Herz spaces, Mediterr. J. Math. 10(4) (2013), 2007–2025. https://doi.org/10.1007/s00009-013-0335-4
• Sultan, B., Sultan, M., Mehmood, M., Azmi, F., Alghafli, M.A., Mlaiki, N., Boundedness of fractional integrals on grand weighted Herz spaces with variable exponent, AIMS Mathematics, 8(1) (2023), 752–764. https://doi.org/10.3934/math.2023036
• Sultan, B, Azmi, F.M., Sultan, M., Mahmood, T., Mlaiki, N., Souayah, N., Boundedness of fractional integrals on grand weighted Herz-Morrey spaces with variable exponent, Fractal and Fractional, 6(11) (2022), 660. https://doi.org/10.3390/fractalfract6110660
• Sultan, B., Azmi, F., Sultan, M., Mehmood, M., Mlaiki, N., Boundedness of Riesz potential operator on grand Herz-Morrey spaces, Axioms, 11(11) (2022), 583. https://doi.org/10.3390/axioms11110583
• Sultan, M., Sultan, B., Aloqaily, A., Mlaiki, N., Boundedness of some operators on grand Herz spaces with variable exponent, AIMS Mathematics, 8(6) (2023), 12964–12985. https://doi.org/10.3934/math.2023653
• Sultan, B., Sultan, M., Khan, A., Abdeljawad, T., Boundedness of Marcinkiewicz integral operator of variable order in grand Herz-Morrey spaces, AIMS Mathematics, 8(9) (2023), 22338–22353. https://doi.org/10.3934/math.20231139
• Sultan, B., Sultan, M., Zhang, Q. Q., Mlaiki, N., Boundedness of Hardy operators on grand variable weighted Herz spaces, AIMS Mathematics, 8(10) (2023), 24515–24527.
• Sultan, B., Sultan, M., Khan, I., On Sobolev theorem for higher commutators of fractional integrals in grand variable Herz spaces, Commun. Nonlinear Sci. Numer. Simul., 126 (2023). DOI 10.1016/j.cnsns.2023.107464
• Sultan, B., Sultan, M., Boundedness of commutators of rough Hardy operators on grand variable Herz spaces, Forum Mathematicum, 2023. https://doi.org/10.1515/forum-2023-0152
• Uribe, D. C., Fiorenza, A., Variable Lebesgue spaces, Foundations and Harmonic Analysis, Appl. Numer. Harmon. Anal., Birkhäuser, Heidelberg, 2013.
• Uribe, D. C., et al., The boundedness of classical operators on variable $L^{p}$ spaces, Acad. Sci. Fenn. Math., 31(1) (2006), 239–264.
• Wang, H., Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent, Czech Math J., 66 (2016), 251–269.