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\(L\log L\) Type Estimates for Commutators of Fractional Integral Operators on the p-Adic Vector Space

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Abstract

In this paper, the main aim is to prove the weak type \(L \log L\) estimates for commutators of fractional integral operators and the higher order in the context of the p-adic version of Lebesgue spaces, where the symbols of the commutators belong to the p-adic version of \({\text {BMO}}\) space. In addition, we also establish the estimates of the sharp function on the p-adic vector space.

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References

  1. Albeverio, S., Khrennikov, A.Y., Shelkovich, V.M.: Harmonic analysis in the \(p\)-adic Lizorkin spaces: fractional operators, pseudo-differential equations, \(p\)-adic wavelets, Tauberian theorems. J. Fourier Anal. Appl. 12, 393–425 (2006)

    Article  MathSciNet  Google Scholar 

  2. Khrennikov, A.Y., Shelkovich, V.M.: Non-Haar \(p\)-adic wavelets and their application to pseudo-differential operators and equations. Appl. Comput. Harmon. Anal. 28(1), 1–23 (2010)

    Article  MathSciNet  Google Scholar 

  3. Torresblanca-Badillo, A., Albarracín-Mantilla, A.A.: Some further classes of pseudo-differential operators in the \(p\)-adic context and their applications. J. Pseudo-Differ. Oper. Appl. 14(2), 24 (2023)

    Article  MathSciNet  Google Scholar 

  4. Shelkovich, V., Skopina, M.: \(p\)-Adic Haar multiresolution analysis and pseudo-differential operators. J. Fourier Anal. Appl. 15, 366–393 (2009)

    Article  MathSciNet  Google Scholar 

  5. Volovich, I.V., Zelenov, E.I.: p-Adic Analysis and Mathematical Physics, vol. 1. World Scientific, Singapore (1994)

    Google Scholar 

  6. Taibleson, H.M.: Fourier Analysis on Local Field. Princeton University Press, Princeton (1975)

    Google Scholar 

  7. Rashid, S., Chu, Y.M., Singh, J., Kumar, D.: A unifying computational framework for novel estimates involving discrete fractional calculus approaches. Alex. Eng. J. 60(2), 2677–2685 (2021)

    Article  Google Scholar 

  8. Zhou, S.S., Rashid, S., Parveen, S., Akdemir, A.O., Hammouch, Z.: New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operators. AIMS Math. 6(5), 4507–4525 (2021)

    Article  MathSciNet  Google Scholar 

  9. Rashid, S., Khalid, A., Sultana, S., Hammouch, Z., Shah, R., Alsharif, A.M.: A novel analytical view of time-fractional Korteweg–De Vries equations via a new integral transform. Symmetry 13(7), 1254 (2021)

    Article  Google Scholar 

  10. Abdeljawad, T., Rashid, S., Hammouch, Z., İ İşcan., Chu, Y.M.: Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications. Adv. Differ. Equ. 2020(1), 1–26 (2020)

  11. Ge-JiLe, H., Rashid, S., Noor, M.A., Suhail, A., Chu, Y.M.: Some unified bounds for exponentially tgs-convex functions governed by conformable. AIMS Math. 5(6), 6108–6123 (2020)

    Article  MathSciNet  Google Scholar 

  12. Wang, H.: Estimates for fractional integral operators and linear commutators on certain weighted amalgam spaces. J. Funct. Spaces 2020, 1–25 (2020)

    MathSciNet  Google Scholar 

  13. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions, vol. 2. Princeton University Press, Princeton (1970)

    Google Scholar 

  14. Haran, S.: Riesz potentials and explicit sums in arithmetic. Invent. Math. 101(1), 697–703 (1990)

    Article  MathSciNet  Google Scholar 

  15. Haran, S.: Analytic potential theory over the \(p\)-adics. Ann. Iinst. Fourier 43(4), 905–944 (1993)

    Article  MathSciNet  Google Scholar 

  16. Sarfraz, N., Aslam, M., Zaman, M., Jarad, F.: Estimates for p-adic fractional integral operator and its commutators on \(p\)-adic Morrey–Herz spaces. J. Inequal. Appl. 2022(1), 1–17 (2022)

    Article  MathSciNet  Google Scholar 

  17. Mo, H.X., Wang, X.J., Ma, R.Q.: Commutator of Riesz potential in \(p\)-adic generalized Morrey spaces. Front. Math. China 13, 633–645 (2018)

    Article  MathSciNet  Google Scholar 

  18. Wu, Q.Y., Fu, Z.W.: Hardy–Littlewood–Sobolev inequalities on \(p\)-adic central Morrey spaces. J. Funct. Spaces 2015 (2015)

  19. Volosivets, S.S.: Generalized fractional integrals in \(p\)-adic Morrey and Herz spaces. p-Adic Numbers Ultrametric Anal. Appl. 9, 53–61 (2017)

  20. Ding, Y., Lu, S., Zhang, P.: Weak estimates for commutators of fractional integral operators. Sci. China Ser. A. 31, 877–888 (2001)

    Article  MathSciNet  Google Scholar 

  21. Cruz-Uribe, Z., Fiorenza, A.: Endpoint estimates and weighted norm inequalities for commutators of fractional integrals. Publ. Mat. 103–131 (2003)

  22. Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces. M. Dekker (1991)

  23. Kim, Y.C.: Carleson measures and the BMO space on the \(p\)-adic vector space. Math. Nachr. 282(9), 1278–1304 (2009)

    Article  MathSciNet  Google Scholar 

  24. He, Q.J., Li, X.: Characterization of Lipschitz spaces via commutators of maximal function on the \(p\)-Adic vector space. J. Math. 2022 (2022)

  25. He, Q.J., Li, X.: Necessary and sufficient conditions for boundedness of commutators of maximal function on the \(p\)-adic vector spaces. AIMS Math. 8(6), 14064–14085 (2023)

    Article  MathSciNet  Google Scholar 

  26. Wu, J.L., Chang, Y.P.: Characterization of boundedness of some commutators of fractional maximal functions in terms of \(p\)-adic vector spaces. arXiv:2308.02736

  27. García-Cuerva, J., De Francia, J.R.: Weighted Norm Inequalities and Related Topics. Elsevier (2011)

  28. Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42(1), 765–778 (1975)

    MathSciNet  Google Scholar 

  29. Wu, Q.Y.: Boundedness for commutators of fractional \(p\)-adic Hardy operators. J. Inequal. Appl. 2012(1), 1–12 (2012)

    Article  MathSciNet  Google Scholar 

  30. Kim, Y.C.: \(L^q\)-estimates of maximal operators on the \(p\)-adic vector space. Commun. Korean Soc. 24(3), 367–379 (2009)

    Article  MathSciNet  Google Scholar 

  31. Grafakos, L.: Modern Fourier analysis. In: Graduate Texts in Mathematics, vol. 250 (2014)

  32. Zheng, S.J., Su, W.: \(L^{p}\) estimates for the iterated Hardy–Littlewood maximal operator on (\(R^{n}\)) and \(K^{n}\), k a local field and \(K^{n}\), k a local field. 14(3), 36–54 (1998)

  33. Pérez, C.: On sufficient conditions for the boundedness of the Hardy–Littlewood maximal operator between weighted \(L^{p}\)-spaces with different weights. Proc. Lond. Math. Soc. 71(1), 135–157 (1995)

    Article  Google Scholar 

  34. Stein, E.: Note on the class LlogL. Studia Math. 3(32), 305–310 (1969)

    Article  Google Scholar 

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Funding

This work was partly supported by the Fundamental Research Funds for Education Department of Heilongjiang Province (Nos. 1453ZD031, 2019-KYYWF-0909 and SJGY20220609) the Reform and Development Foundation for Local Colleges and Universities of the Central Government (No. 2020YQ07), the MNU (Nos. KCSZKC-2022026 and KCSZAL-2022013) and Mudanjiang Normal University Science and Technology Innovation Project (Nos. kjcx2021-056mdjnu and kjcx2023-126mdjnu).

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  1. Department of Mathematics, Mudanjiang Normal University, Mudanjiang, 157011, China

    YunPeng Chang, LiangJuan Yu, LinQi Sun & HuangZhi Xia

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  1. YunPeng Chang
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All authors contributed equally to the writing of this article. All authors read the final manuscript and approved its submission.

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Correspondence to YunPeng Chang.

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Communicated by Tao Qian.

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Chang, Y., Yu, L., Sun, L. et al. \(L\log L\) Type Estimates for Commutators of Fractional Integral Operators on the p-Adic Vector Space. Complex Anal. Oper. Theory 18, 79 (2024). https://doi.org/10.1007/s11785-024-01514-4

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