Commutators of Hardy-Cesàro operators on Morrey-Herz spaces with variable exponents

Kieu Huu Dung; Do Lu Cong Minh; Pham Thi Kim Thuy; Kieu Huu Dung; Do Lu Cong Minh; Pham Thi Kim Thuy

doi:10.3934/math.20221051

AIMS Mathematics

2022, Volume 7, Issue 10: 19147-19166. doi: 10.3934/math.20221051

Previous Article Next Article

Research article

Commutators of Hardy-Cesàro operators on Morrey-Herz spaces with variable exponents

1.
Faculty of Basic Sciences, Van Lang University, Ho Chi Minh City, Vietnam
2.
Department of Mathematics, FPT University, Ho Chi Minh City, Vietnam

Received: 23 May 2022 Revised: 15 August 2022 Accepted: 22 August 2022 Published: 29 August 2022
MSC : 26D10, 42B35

The aim of this paper is to establish some sufficient conditions for the boundedness of commutators of Hardy-Cesàro operators with symbols in central BMO spaces with variable exponent on some function spaces such as the local central Morrey, Herz, and Morrey-Herz spaces with variable exponents.
- Hardy-Cesàro operator,
- commutator,
- local central Morrey space,
- Herz space,
- Morrey-Herz space,
- block space,
- variable exponent
Citation: Kieu Huu Dung, Do Lu Cong Minh, Pham Thi Kim Thuy. Commutators of Hardy-Cesàro operators on Morrey-Herz spaces with variable exponents[J]. AIMS Mathematics, 2022, 7(10): 19147-19166. doi: 10.3934/math.20221051

Related Papers:

Abstract

The aim of this paper is to establish some sufficient conditions for the boundedness of commutators of Hardy-Cesàro operators with symbols in central BMO spaces with variable exponent on some function spaces such as the local central Morrey, Herz, and Morrey-Herz spaces with variable exponents.

References

[1]	A. Almeida, D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl., 394 (2012), 781–795. https://doi.org/10.1016/j.jmaa.2012.04.043 doi: 10.1016/j.jmaa.2012.04.043
[2]	C. Capone, D. Cruz-Uribe, A. Fiorenza, The fractional maximal operator and fractional integrals on variable $L_p$ spaces, Rev. Mat. Iberoam., 23 (2007), 743–770. https://doi.org/10.4171/RMI/511 doi: 10.4171/RMI/511
[3]	C. Carton-Lebrun, M. Fosset, Moyennes et quotients de Taylor dans BMO, Bull. Soc. Roy. Sci. Liége, 53 (1984), 85–87.
[4]	D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue spaces: Foundations and harmonic analysis, Basel: Springer, 2013. https://doi.org/10.1007/978-3-0348-0548-3
[5]	N. M. Chuong, D. V. Duong, K. H. Dung, Multilinear Hausdorff operator on variable exponent Morrey-Herz type spaces, Integr. Transf. Spec. F., 31 (2020), 62–86. https://doi.org/10.1080/10652469.2019.1666375 doi: 10.1080/10652469.2019.1666375
[6]	N. M. Chuong, D. V. Duong, K. H. Dung, Some estimates for $p$-adic rough multilinear Hausdorff operators and commutators on weighted Morrey-Herz type spaces, Russ. J. Math. Phys., 26 (2019), 9–31. https://doi.org/10.1134/S1061920819010023 doi: 10.1134/S1061920819010023
[7]	N. M. Chuong, D. V. Duong, H. D. Hung, Bounds for the weighted Hardy-Cesàro operator and its commutator on weighted Morrey-Herz type spaces, Z. Anal. Anwend., 35 (2016), 489–504. https://doi.org/10.4171/ZAA/1575 doi: 10.4171/ZAA/1575
[8]	N. M. Chuong, H. D. Hung, Bounds of weighted Hardy-Cesàro operators on weighted Lebesgue and BMO spaces, Integr. Transf. Spec. F., 25 (2014), 697–710. https://doi.org/10.1080/10652469.2014.898635 doi: 10.1080/10652469.2014.898635
[9]	D. V. Duong, K. H. Dung, N. M. Chuong, Weighted estimates for commutators of multilinear Hausdorff operators on variable exponent Morrey-Herz type spaces, Czech. Math. J., 70 (2020), 833–865. https://doi.org/10.21136/CMJ.2020.0566-18 doi: 10.21136/CMJ.2020.0566-18
[10]	K. H. Dung, D. V. Duong, T. N. Luan, Weighted central BMO type space estimates for commutators of $p$-adic Hardy-Cesàro operators, P-Adic Num. Ultrametr. Anal. Appl., 13 (2021), 266–279. https://doi.org/10.1134/S2070046621040026 doi: 10.1134/S2070046621040026
[11]	K. H. Dung, P. T. K. Thuy, Commutators of Hausdorff operators on Herz-type Hardy spaces, Adv. Oper. Theory, 7 (2022), 37. https://doi.org/10.1007/s43036-022-00202-4 doi: 10.1007/s43036-022-00202-4
[12]	L. Diening, M. Ružička, Calderón-Zygmund operators on generalized Lebesgue spaces $L^{p(x)}$ and problems related to fluid dynamics, J. Reine Angew. Math., 2003 (2003), 197–220. https://doi.org/10.1515/crll.2003.081 doi: 10.1515/crll.2003.081
[13]	L. Diening, P. Harjulehto, P. Hästö, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Springer, 2011. https://doi.org/10.1007/978-3-642-18363-8
[14]	P. Federbush, Navier and Stokes meet the wavelet, Commun. Math. Phys., 155 (1993), 219–248.
[15]	Z. W. Fu, Z. G. Liu, S. Z. Lu, Commutators of weighted Hardy operators on $\mathbb R^n$, Proc. Amer. Math. Soc., 137 (2009), 3319–3328.
[16]	Z. W. Fu, S. L. Gong, S. Z. Lu, W. Yuan, Weighted multilinear Hardy operators and commutators, Forum Math., 27 (2015), 2825–2851. https://doi.org/10.1515/forum-2013-0064 doi: 10.1515/forum-2013-0064
[17]	L. Grafakos, Modern Fourier analysis, New York: Springer, 2008. https://doi.org/10.1007/978-0-387-09434-2
[18]	H. D. Hung, L. D. Ky, New weighted multilinear operators and commutators of Hardy-Cesàro type, Acta Math. Sci., 35 (2015), 1411–1425. https://doi.org/10.1016/S0252-9602(15)30063-1 doi: 10.1016/S0252-9602(15)30063-1
[19]	K. P. Ho, Fractional geometrical maximal functions on Morrey spaces with variable exponents, Results Math., 77 (2022), 32. https://doi.org/10.1007/s00025-021-01570-8 doi: 10.1007/s00025-021-01570-8
[20]	M. Izuki, Fractional integrals on Herz-Morrey spaces with variable exponent, Hiroshima Math. J., 40 (2010), 343–355. https://doi.org/10.32917/hmj/1291818849 doi: 10.32917/hmj/1291818849
[21]	M. Izuki, T. Noi, Two weighted Herz spaces with variable exponents, Bull. Malays. Math. Sci. Soc., 43 (2020), 169–200. https://doi.org/10.1007/s40840-018-0671-4 doi: 10.1007/s40840-018-0671-4
[22]	S. Lu, L. Xu, Boundedness of rough singular integral operators on the homogeneous Morrey–Herz spaces, Hokkaido Math. J., 34 (2005), 299–313. https://doi.org/10.14492/hokmj/1285766224 doi: 10.14492/hokmj/1285766224
[23]	S. Lu, D. Yang, Some new Hardy spaces associated with Herz spaces and their wavelet characterization, J. Beijing Normal Univ. (Nat. Sci.), 29 (1993), 10–19.
[24]	S. Lu, D. Yang, G. Hu, Herz type spaces and their applications, Beijing: Science Press, 2008.
[25]	Y. Lu, Y. P. Zhu, Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents, Acta Math. Sin., 30 (2014), 1180–1194. https://doi.org/10.1007/s10114-014-3410-2 doi: 10.1007/s10114-014-3410-2
[26]	F. I. Mamedov, A. Harman, On a Hardy type general weighted inequality in spaces $L^{p(\cdot)}$, Integr. Equ. Oper. Theory, 66 (2010), 565–592. https://doi.org/10.1007/s00020-010-1765-z doi: 10.1007/s00020-010-1765-z
[27]	J. Ruan, D. Fan, Q. Wu, Weighted Morrey estimates for Hausdorff operator and its commutator on the Heisenberg group, Math. Inequal. Appl., 22 (2019), 307–329. https://doi.org/10.7153/mia-2019-22-24 doi: 10.7153/mia-2019-22-24
[28]	C. Tang, F. Xue, Y. Zhou, Commutators of weighted Hardy operators on Herz-type spaces, Ann. Pol. Math., 101 (2011), 267–273. https://doi.org/10.4064/ap101-3-6 doi: 10.4064/ap101-3-6
[29]	M. E. Taylor, Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations, Commun. Part. Diff. Eq., 17 (1992), 1407–1456. https://doi.org/10.1080/03605309208820892 doi: 10.1080/03605309208820892
[30]	D. H. Wang, Z. G. Liu, J. Zhou, Z. D. Teng, Central BMO spaces with variable exponent, 2018, arXiv: 1708.00285.
[31]	H. Wang, Anisotropic Herz spaces with variable exponents, Commun. Math. Anal., 18 (2015), 1–14.
[32]	J. L. Wu, W. J. 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 doi: 10.1215/21562261-3664932
[33]	L. W. Wang, L. S. Shu, Higher order commutators of fractional integrals on Morrey type spaces with variable exponents, Math. Nachr., 291 (2018), 1437–1449. https://doi.org/10.1002/mana.201600438 doi: 10.1002/mana.201600438
[34]	B. Xu, Bilinear $\theta$-type Calderón-Zygmund operators and its commutators on generalized variable exponent Morrey spaces, AIMS Math., 7 (2022), 12123–12143. https://doi.org/10.3934/math.2022674 doi: 10.3934/math.2022674
[35]	J. Xiao, $L^p$ and $BMO$ bounds of weighted Hardy-Littlewood averages, J. Math. Anal. Appl., 262 (2001), 660–666. https://doi.org/10.1006/jmaa.2001.7594 doi: 10.1006/jmaa.2001.7594
[36]	Y. Zhu, Y. Tang, L. Jiang, Boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and Morrey-Herz spaces with variable exponents, AIMS Math., 6 (2021), 11246–11262. https://doi.org/10.3934/math.2021652 doi: 10.3934/math.2021652

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)