VARIATION INEQUALITIES FOR THE COMMUTATORS OF APPROXIMATE IDENTITIES WITH LIPSCHITZ FUNCTIONS

Wuxin Liu; Jing Zhang

doi:10.11948/20250103

2026 Volume 16 Issue 1

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Wuxin Liu, Jing Zhang. VARIATION INEQUALITIES FOR THE COMMUTATORS OF APPROXIMATE IDENTITIES WITH LIPSCHITZ FUNCTIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(1): 155-172. doi: 10.11948/20250103

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Wuxin Liu, Jing Zhang. VARIATION INEQUALITIES FOR THE COMMUTATORS OF APPROXIMATE IDENTITIES WITH LIPSCHITZ FUNCTIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(1): 155-172. doi: 10.11948/20250103

VARIATION INEQUALITIES FOR THE COMMUTATORS OF APPROXIMATE IDENTITIES WITH LIPSCHITZ FUNCTIONS

Wuxin Liu^1,,
Jing Zhang^1,2, ,

1.
School of Mathematics and Statistics, Yili Normal University, 835000 Yining, China
2.
Institute of Applied Mathematics, Yili Normal University, 835000 Yining, China

Author Bio: Email: 15294456309@163.com(W. Liu)
Corresponding author: Email: jzhang@ylnu.edu.cn(J. Zhang)
Fund Project: The authors were supported by National Natural Science Foundation of China (12361019), Xinjiang Uygur Autonomous Region Graduate Innovation Project (Grant No. XJ2023G259)

Abstract

Abstract

This paper is devoted to establishing the boundedness of the variation operators for commutators generated by approximate identities with Lipschitz functions in the weighted Lebesgue spaces and the endpoint spaces. As applications, we obtain the corresponding boundedness results for $ \lambda $-jump operator, the number of up-crossing, heat semigroups, Poisson semigroups and maximal operator.
- Variation operators /
- approximate identities /
- commutators /
- Lipschitz functions /
- boundedness
MSC: 42B20, 42B25, 42B30, 42B35

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References

[1]	A. Barbagallo and M. A. Ragusa, Preface of “The 2nd symposium on variational inequalities and equilibrium problems”, AIP Conf. Proc., 2010, 1281(1), 259–260. Google Scholar
[2]	J. J. Betancor and M. De León-Contreras, Variation inequalities for Riesz transforms and Poisson semigroups associated with Laguerre polynomial expansions, Anal. Appl., 2024, 26(1), 1–52. Google Scholar
[3]	J. J. Betancor, J. C. Fariña, E. Harboure and L. Rodríguez-Mesa, L^p-boundedness properties of variation operators in the Schrödinger setting, Rev. Mat. Complut., 2013, 26(2), 485–534. Google Scholar
[4]	J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Publ. Math. Inst. Hautes Études Sci., 1989, 69(1), 5–45. Google Scholar
[5]	J. T. Campbell, R. L. Jones, K. Reinhold and M. Wierdl, Oscillations and variation for the Hilbert transform, Duke Math. J., 2000, 105(1), 59–83. Google Scholar
[6]	J. T. Campbell, R. L. Jones, K. Reinhold and M. Wierdl, Oscillations and variation for singular integrals in higher dimensions, Trans. Amer. Math. Soc., 2003, 355(5), 2115–2137. Google Scholar
[7]	S. Chen, H. Wu and Q. Xue, A note on multilinear Muckenhoupt classes for multiple weights, Studia Math., 2014, 223(1), 1–18. Google Scholar
[8]	Y. Chen, L. Yang and M. Qu, Weighted norm inequalities for jump and variation of some singular integral operators with rough kernel, Internat. J. Math., 2025, 36(5), 2450088. Google Scholar
[9]	J. García-Cuerva, Weighted H^p Space, Instytut Matematyczny Polskiej Akademi Nauk, Warszawa, 1979. Google Scholar
[10]	S. Demir, Variation and $\lambda$-jump inequalities on H^p Spaces, Russian Math., 2024, 68(4), 12–16. $\lambda$-jump inequalities on H^p Spaces" target="_blank">Google Scholar
[11]	X. T. Duong, J. Li and D. Yang, Variation of Calderón-Zygmund operators with matrix weight, Commun. Contemp. Math., 2021, 23(7), 2050062. Google Scholar
[12]	M. Eslamian, Variational inequality problem over the solution set of split monotone variational inclusion problem with application to bilevel programming problem, Filomat, 2024, 37(24), 8361–8376. Google Scholar
[13]	U. A. Ezeafulukwe, G. B. Akuchu, S. Etemad, A. E. Ofem, G. C. Ugwunnadi and Z. M. Yaseen, On the monotone variational inclusion problems: A new algorithm-based modiffed splitting approach, J. Funct. Spaces, 2025, 2025(1), 7233178. Google Scholar
[14]	C. Fefferman and E. M. Stein, H^p spaces of several variables, Acta Math., 1972, 129(1), 137–193. Google Scholar
[15]	W. Guo, Y. Wen, H. Wu and D. Yang, Variational characterizations of weighted Hardy spaces and weighted BMO spaces, Proc. Roy. Soc. Edinburgh Sect. A, 2022, 152(6), 1613–1632. Google Scholar
[16]	R. L. Jones, Variation inequalities for singular integrals and related operators, Contemp. Math., 2006, 411, 89–121. Google Scholar
[17]	D. Lépingle, La variation d'order $p$ des semi-martingales, Z. Wahrsch. Verw. Geb., 1976, 36(4), 295–316. $p$ des semi-martingales" target="_blank">Google Scholar
[18]	F. Liu and H. Wu, A criterion on oscillation and variation for the commutators of singular integral operators, Forum Math., 2015, 27(1), 77–97. Google Scholar
[19]	H. Liu, Variational characterization of H^p, Proc. Roy. Soc. Edinburgh Sect. A, 2019, 149(5), 1123–1134. Google Scholar
[20]	B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 1972, 165, 207–226. Google Scholar
[21]	B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the fractional integrals, Trans. Amer. Math. Soc., 1974, 192, 261–274. Google Scholar
[22]	Z. Si and L. Wang, Weighted inequalities for commutators associated with multilinear maximal square functions, Front. Math., 2023, 18(6), 1315–1330. Google Scholar
[23]	E. Stein and T. S. Murphy, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ., 1993. Google Scholar
[24]	Y. Wen and X. Hou, The boundedness of variation associated with the commutators of approximate identities, J. Inequal. Appl., 2021, 2021(1), 140. Google Scholar
[25]	G. Wu, L. Tang and L. Yang, Weighted estimates for pseudo-differential operators with weak regular symbols and their commutators, J. Pseudo-Differ. Oper. Appl., 2025, 16(1), 7. Google Scholar
[26]	H. Wu, D. Yang and J. Zhang, Oscillation and variation for semigroups associated with Bessel operators, J. Math. Anal. Appl., 2016, 443(2), 848–867. Google Scholar
[27]	H. Wu, D. Yang and J. Zhang, Oscillation and variation for the Riesz transform associated with Bessel operators, Proc. Roy. Soc. Edinburgh Sect. A, 2019, 149(1), 169–190. Google Scholar
[28]	J. Zhang and H. Wu, Oscillation and variation inequalities for the commutators of singular integrals with Lipschitz functions, J. Inequal. Appl., 2015, 2015(1), 214. Google Scholar