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A NOTE ON VARIATION CONTINUITY FOR MULTILINEAR MAXIMAL OPERATORS

  • Xiao Zhang (College of Electronic and Information Engineering Shandong University of Science and Technology)
  • Received : 2023.02.12
  • Accepted : 2023.04.21
  • Published : 2024.01.31
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Abstract

This note is devoted to establishing the variation continuity of the one-dimensional discrete uncentered multilinear maximal operator. The above result is based on some refine variation estimates of the above maximal functions on monotone intervals. The main result essentially improves some known ones.

Keywords

Acknowledgement

The author wants to express her sincere thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable.

References

  1. J. M. Aldaz and F. Perez Lazaro, Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2443-2461. https://doi.org/10.1090/S0002-9947-06-04347-9
  2. D. Beltran and J. Madrid, Regularity of the centered fractional maximal function on radial functions, J. Funct. Anal. 279 (2020), no. 8, 108686, 28 pp. https://doi.org/10.1016/j.jfa.2020.108686
  3. J. W. Bober, E. Carneiro, K. Hughes, and L. B. Pierce, On a discrete version of Tanaka's theorem for maximal functions, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1669-1680. https://doi.org/10.1090/S0002-9939-2011-11008-6
  4. H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486-490. https://doi.org/10.2307/2044999
  5. E. Carneiro and K. Hughes, On the endpoint regularity of discrete maximal operators, Math. Res. Lett. 19 (2012), no. 6, 1245-1262. https://doi.org/10.4310/MRL.2012.v19.n6.a6
  6. E. Carneiro and J. Madrid, Derivative bounds for fractional maximal functions, Trans. Amer. Math. Soc. 369 (2017), no. 6, 4063-4092. https://doi.org/10.1090/tran/6844
  7. E. Carneiro, J. Madrid, and L. B. Pierce, Endpoint Sobolev and BV continuity for maximal operators, J. Funct. Anal. 273 (2017), no. 10, 3262-3294. https://doi.org/10.1016/j.jfa.2017.08.012
  8. E. Carneiro and D. Moreira, On the regularity of maximal operators, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4395-4404. https://doi.org/10.1090/S0002-9939-08-09515-4
  9. E. Carneiro and B. F. Svaiter, On the variation of maximal operators of convolution type, J. Funct. Anal. 265 (2013), no. 5, 837-865. https://doi.org/10.1016/j.jfa.2013.05.012
  10. B. Dong, Z. Fu, and J. Xu, Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations, Sci. China Math. 61 (2018), no. 10, 1807-1824. https://doi.org/10.1007/s11425-017-9274-0
  11. C. Gonz'alez-Riquelme and D. Kosz, BV continuity for the uncentered Hardy-Littlewood maximal operator, J. Funct. Anal. 281 (2021), no. 2, Paper No. 109037, 20 pp. https://doi.org/10.1016/j.jfa.2021.109037
  12. P. Haj lasz and J. Onninen, On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 1, 167-176.
  13. J. Kinnunen, The Hardy-Littlewood maximal function of a Sobolev function, Israel J. Math. 100 (1997), 117-124. https://doi.org/10.1007/BF02773636
  14. J. Kinnunen and P. Lindqvist, The derivative of the maximal function, J. Reine Angew. Math. 503 (1998), 161-167. https://doi.org/10.1515/crll.1998.095
  15. J. Kinnunen and E. Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), no. 4, 529-535. https://doi.org/10.1112/S0024609303002017
  16. O. Kurka, On the variation of the Hardy-Littlewood maximal function, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 109-133. https://doi.org/10.5186/aasfm.2015.4003
  17. F. Liu, T. Chen, and H. Wu, A note on the endpoint regularity of the Hardy-Littlewood maximal functions, Bull. Aust. Math. Soc. 94 (2016), no. 1, 121-130. https://doi.org/10.1017/S0004972715001392
  18. F. Liu and H. Wu, Endpoint regularity of multisublinear fractional maximal functions, Canad. Math. Bull. 60 (2017), no. 3, 586-603. https://doi.org/10.4153/CMB-2016-044-9
  19. F. Liu and H. Wu, Regularity of discrete multisublinear fractional maximal functions, Sci. China Math. 60 (2017), no. 8, 1461-1476. https://doi.org/10.1007/s11425-016-9011-2
  20. H. Luiro, Continuity of the maximal operator in Sobolev spaces, Proc. Amer. Math. Soc. 135 (2007), no. 1, 243-251. https://doi.org/10.1090/S0002-9939-06-08455-3
  21. H. Luiro, On the regularity of the Hardy-Littlewood maximal operator on subdomains of Rn, Proc. Edinb. Math. Soc. (2) 53 (2010), no. 1, 211-237. https://doi.org/10.1017/S0013091507000867
  22. J. Madrid, Sharp inequalities for the variation of the discrete maximal function, Bull. Aust. Math. Soc. 95 (2017), no. 1, 94-107. https://doi.org/10.1017/S0004972716000903
  23. S. Shi, Z. Fu, and S. Z. Lu, On the compactness of commutators of Hardy operators, Pacific J. Math. 307 (2020), no. 1, 239-256. https://doi.org/10.2140/pjm.2020.307.239
  24. S. Shi and J. Xiao, On fractional capacities relative to bounded open Lipschitz sets, Potential Anal. 45 (2016), no. 2, 261-298. https://doi.org/10.1007/s11118-016-9545-2
  25. S. Shi, L. Zhang, and G. Wang, Fractional non-linear regularity, potential and balayage, J. Geom. Anal. 32 (2022), no. 8, Paper No. 221, 29 pp. https://doi.org/10.1007/s12220-022-00956-6
  26. H. Tanaka, A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function, Bull. Austral. Math. Soc. 65 (2002), no. 2, 253-258. https://doi.org/10.1017/S0004972700020293
  27. F. Temur, On regularity of the discrete Hardy-Littlewood maximal function, preprint at http://arxiv.org/abs/1303.3993.
  28. S. Yang, D. Chang, D. Yang, and Z. Fu, Gradient estimates via rearrangements for solutions of some Schrodinger equations, Anal. Appl. (Singap.) 16 (2018), no. 3, 339-361. https://doi.org/10.1142/S0219530517500142
  29. M. Yang, Z. Fu, and S. Liu, Analyticity and existence of the Keller-Segel-Navier-Stokes equations in critical Besov spaces, Adv. Nonlinear Stud. 18 (2018), no. 3, 517-535. https://doi.org/10.1515/ans-2017-6046
  30. M. Yang, Z. Fu, and J. Sun, Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces, Sci. China Math. 60 (2017), no. 10, 1837-1856. https://doi.org/10.1007/s11425-016-0490-y
  31. M. Yang, Z. Fu, and J. Sun, Existence and large time behavior to coupled chemotaxis-fluid equations in Besov-Morrey spaces, J. Differential Equations 266 (2019), no. 9, 5867-5894. https://doi.org/10.1016/j.jde.2018.10.050
  32. X. Zhang, Endpoint regularity of the discrete multisublinear fractional maximal operators, Results Math. 76 (2021), no. 2, Paper No. 77, 21 pp. https://doi.org/10.1007/s00025-021-01387-5