Boundedness of Hardy operators on grand variable weighted Herz spaces

Babar Sultan; Mehvish Sultan; Qian-Qian Zhang; Nabil Mlaiki; Babar Sultan; Mehvish Sultan; Qian-Qian Zhang; Nabil Mlaiki

doi:10.3934/math.20231250

AIMS Mathematics

2023, Volume 8, Issue 10: 24515-24527. doi: 10.3934/math.20231250

Previous Article Next Article

Research article

Boundedness of Hardy operators on grand variable weighted Herz spaces

1.
Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan
2.
Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan
3.
College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China
4.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia

Received: 06 July 2023 Revised: 08 August 2023 Accepted: 13 August 2023 Published: 18 August 2023
MSC : 46E30, 47B38

In this paper, we will introduce the idea of grand variable weighted Herz spaces $ {{\dot{K} ^{\alpha(\cdot), \epsilon), \theta}_{ q(\cdot)}(\tau)}} $ in which $ \alpha $ is also a variable. Our main purpose in this paper is to prove the boundedness of Hardy operators on grand variable weighted Herz spaces.
- Hardy operators,
- grand Herz spaces,
- weighted Herz spaces,
- grand weighted Herz spaces
Citation: Babar Sultan, Mehvish Sultan, Qian-Qian Zhang, Nabil Mlaiki. Boundedness of Hardy operators on grand variable weighted Herz spaces[J]. AIMS Mathematics, 2023, 8(10): 24515-24527. doi: 10.3934/math.20231250

Related Papers:

Abstract

In this paper, we will introduce the idea of grand variable weighted Herz spaces $ {{\dot{K} ^{\alpha(\cdot), \epsilon), \theta}_{ q(\cdot)}(\tau)}} $ in which $ \alpha $ is also a variable. Our main purpose in this paper is to prove the boundedness of Hardy operators on grand variable weighted Herz spaces.

References

[1]	C. S. Herz, Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms, J. Math. Mech., 18 (1968), 283–323.
[2]	S. Shi, Z. Fu, S. Lu, On the compactness of commutators of Hardy operators, Pac. J. Math., 1 (2020), 239–256. https://doi.org/10.2140/pjm.2020.307.239 doi: 10.2140/pjm.2020.307.239
[3]	Z. Fu, S. Lu, S. Shi, Two characterizations of central BMO space via the commutators of Hardy operators, Forum Math., 33 (2021), 505–529. https://doi.org/10.1515/forum-2020-0243 doi: 10.1515/forum-2020-0243
[4]	S. Lu, Z. Fu, F. Zhao, S. Shi, Hardy operators on Euclidean spaces and ralated topics, Singapore: World Scientific Publishing, 2022.
[5]	D. C. Uribe, A. Fiorenza, Variable Lebesgue space: Foundations and harmonic analysis, Basel: Birkhauser, 2013. https://doi.org/10.1007/978-3-0348-0548-3
[6]	V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral operators in non-standard function spaces, Springer, 2 (2016).
[7]	M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math., 36 (2010), 33–50. https://doi.org/10.1007/s10476-010-0102-8 doi: 10.1007/s10476-010-0102-8
[8]	M. Izuki, Boundedness of vector-valued sublinear operators on Herz-Morrey spaces with variable exponent, Math. Sci. Res. J., 13 (2009), 243–253.
[9]	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
[10]	M. Izuki, T. Noi, Boundedness of fractional integrals on weighted Herz spaces with variable exponent, J. Inequal. Appl., 199 (2016), 1–15. https://doi.org/10.1186/s13660-016-1142-9 doi: 10.1186/s13660-016-1142-9
[11]	M. Izuki, T. Noi, An intrinsic square function on weighted Herz spaces with variable exponents, J. Math. Inequal., 11 (2017), 799–816. https://doi.org/10.48550/arXiv.1606.01019 doi: 10.48550/arXiv.1606.01019
[12]	A. Meskhi, Maximal functions, potentials and singular integrals in grand Morrey spaces, Complex Var. Elliptic, 2010, 1007–1186. https://doi.org/10.1080/17476933.2010.534793 doi: 10.1080/17476933.2010.534793
[13]	H. Nafis, H. Rafeiro, M. A. Zaighum, A note on the boundedness of sublinear operators on grand variable Herz spaces, J. Inequal. Appl., 2020 (2020), 1–13. https://doi.org/10.1186/s13660-019-2265-6 doi: 10.1186/s13660-019-2265-6
[14]	H. Nafis, H. Rafeiro, M. A. Zaighum, Boundedness of the Marcinkiewicz integral on grand Herz spaces, J. Math. Inequal., 15 (2021), 739–753. https://doi.org/10.7153/jmi-2021-15-52 doi: 10.7153/jmi-2021-15-52
[15]	H. Nafis, H. Rafeiro, M. A. Zaighum, Boundedness of multilinear Calderón-Zygmund operators on grand variable Herz spaces, J. Funct. Space., 2022 (2022).
[16]	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 doi: 10.3934/math.2023653
[17]	S. Bashir, B. Sultan, A. Hussain, A. Khan, T. Abdeljawad, A note on the boundedness of Hardy operators in grand Herz spaces with variable exponent, AIMS Math., 8 (2023), 22178–22191. https://doi.org/10.3934/math.20231130 doi: 10.3934/math.20231130
[18]	L. Wang, Parametrized Littlewood-Paley operators on grand variable Herz spaces, Ann. Funct. Anal., 13 (2022). https://doi.org/10.1007/s43034-022-00218-0 doi: 10.1007/s43034-022-00218-0
[19]	B. Sultan, F. Azmi, M. Sultan, M. Mehmood, N. Mlaiki, Boundedness of Riesz potential operator on grand Herz-Morrey spaces, Axioms, 11 (2022), 583.
[20]	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 doi: 10.3934/math.20231139
[21]	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 doi: 10.3934/math.2023036
[22]	B. Sultan, F. 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), 660–670. https://doi.org/10.3390/fractalfract6110660 doi: 10.3390/fractalfract6110660
[23]	L. Diening, A. Peter, Muckenhoupt weights in variable exponent spaces, Mathematik, 2008.
[24]	A. Hussain, M. Asim, M. Aslam, F. Jarad, Commutators of the fractional Hardy operator on weighted variable Herz-Morrey spaces, J. Funct. Space., 2021 (2021), 10. https://doi.org/10.1155/2021/9705250 doi: 10.1155/2021/9705250
[25]	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 doi: 10.1515/math-2020-0175
[26]	A. Hussain, A. Ajaib, Some results for the commutators of generalized Hausdorff operator, J. Math. Inequal., 13 (2019), 1129–1146. https://doi.org/10.48550/arXiv.1804.05309 doi: 10.48550/arXiv.1804.05309

Reader Comments

Your name:*

Email:*
© 2023 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)