대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제60권5호
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  • Pages.1391-1408
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  • 2023
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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COMMUTATORS OF THE MAXIMAL FUNCTIONS ON BANACH FUNCTION SPACES

  • Mujdat Agcayazi (Department of Mathematics and Science Education Aydin Adnan Menderes University) ;
  • Pu Zhang (Department of Mathematics Mudanjiang Normal University)
  • 투고 : 2022.10.16
  • 심사 : 2023.03.30
  • 발행 : 2023.09.30
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초록

Let M and M# be Hardy-Littlewood maximal operator and sharp maximal operator, respectively. In this article, we present necessary and sufficient conditions for the boundedness properties for commutator operators [M, b] and [M#, b] in a general context of Banach function spaces when b belongs to BMO(?n) spaces. Some applications of the results on weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces and Musielak-Orlicz spaces are also given.

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과제정보

Pu Zhang was supported by National Natural Science Foundation of China (Grant No. 11571160).

참고문헌

  1. M. Agcayazi, A. Gogatishvili, K. Koca, and R. Mustafayev, A note on maximal commutators and commutators of maximal functions, J. Math. Soc. Japan 67 (2015), no. 2, 581-593. https://doi.org/10.2969/jmsj/06720581
  2. J. Bastero, M. Milman, and F. J. Ruiz Blasco, Commutators for the maximal and sharp functions, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3329-3334. https://doi.org/10.1090/S0002-9939-00-05763-4
  3. C. Bennett and R. C. Sharpley, Interpolation of operators, Pure and Applied Mathematics, 129, Academic Press, Inc., Boston, MA, 1988.
  4. A. Bonami, T. Iwaniec, P. Jones, and M. Zinsmeister, On the product of functions in BMO and H1, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 5, 1405-1439. https://doi.org/10.5802/aif.2299
  5. R. R. Coifman and C. L. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. https://doi.org/10.4064/sm-51-3-241-250
  6. R. R. Coifman, R. Rochberg, and G. L. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611-635. https://doi.org/10.2307/1970954
  7. D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue spaces, Applied and Numerical Harmonic Analysis, Birkhauser/Springer, Heidelberg, 2013. https://doi.org/10.1007/978-3-0348-0548-3
  8. D. V. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, The maximal function on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 223-238.
  9. D. V. Cruz-Uribe and P. A. Hasto, Extrapolation and interpolation in generalized Orlicz spaces, Trans. Amer. Math. Soc. 370 (2018), no. 6, 4323-4349. https://doi.org/10.1090/tran/7155
  10. D. V. Cruz-Uribe, J. Martell, and C. Perez Moreno, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, 215, Birkhauser/Springer Basel AG, Basel, 2011. https://doi.org/10.1007/978-3-0348-0072-3
  11. L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), no. 8, 657-700. https://doi.org/10.1016/j.bulsci.2003.10.003
  12. L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8
  13. X. Fu, D. C. Yang, and W. Yuan, Boundedness of multilinear commutators of Calderon- Zygmund operators on Orlicz spaces over non-homogeneous spaces, Taiwanese J. Math. 16 (2012), no. 6, 2203-2238. https://doi.org/10.11650/twjm/1500406848
  14. J. Garcia-Cuerva, E. Harboure, C. Segovia, and J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J. 40 (1991), no. 4, 1397-1420. https://doi.org/10.1512/iumj.1991.40.40063
  15. J. Garcia-Cuerva and J. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, 116, North-Holland, Amsterdam, 1985.
  16. A. Gogatishvili, R. C. Mustafayev, and M. Agcayazi, Weak-type estimates in Morrey spaces for maximal commutator and commutator of maximal function, Tokyo J. Math. 41 (2018), no. 1, 193-218. https://doi.org/10.3836/tjm/1502179258
  17. P. Harjulehto and P. A. Hasto, Uniform convexity and associate spaces, Czechoslovak Math. J. 68(143) (2018), no. 4, 1011-1020. https://doi.org/10.21136/CMJ.2018.0054-17
  18. P. Harjulehto and P. A. Hasto, Orlicz spaces and generalized Orlicz spaces, Lecture Notes in Mathematics, 2236, Springer, Cham, 2019. https://doi.org/10.1007/978-3-030-15100-3
  19. P. A. Hasto, The maximal operator on generalized Orlicz spaces, J. Funct. Anal. 269 (2015), no. 12, 4038-4048. https://doi.org/10.1016/j.jfa.2015.10.002
  20. G. E. Hu, H. B. Lin, and D. C. Yang, Commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type, Abstr. Appl. Anal. 2008 (2008), Art. ID 237937, 21 pp. https://doi.org/10.1155/2008/237937
  21. G. E. Hu and D. C. Yang, Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type, J. Math. Anal. Appl. 354 (2009), no. 1, 249-262. https://doi.org/10.1016/j.jmaa.2008.12.066
  22. M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math. 36 (2010), no. 1, 33-50. https://doi.org/10.1007/s10476-010-0102-8
  23. M. Izuki, Another proof of characterization of BMO via Banach function spaces, Rev. Un. Mat. Argentina 57 (2016), no. 1, 103-109.
  24. S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), no. 2, 263-270. https://doi.org/10.1007/BF02386000
  25. V. M. Kokilashvili and M. Krbec, Weighted inequalities in Lorentz and Orlicz spaces, World Sci. Publishing, Inc., River Edge, NJ, 1991. https://doi.org/10.1142/9789814360302
  26. M. A. Krasnosel'skii and Y. B. Rutitskii, Convex functions and Orlicz spaces, translated from the first Russian edition by Leo F. Boron, Noordhoff, Groningen, 1961.
  27. A. Kufner, O. John, and S. Fucik, Function spaces, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff, Leyden, 1977.
  28. D. F. Li, G. E. Hu, and X. L. Shi, Weighted norm inequalities for the maximal commutators of singular integral operators, J. Math. Anal. Appl. 319 (2006), no. 2, 509-521. https://doi.org/10.1016/j.jmaa.2005.06.054
  29. M. Milman and T. P. Schonbek, Second order estimates in interpolation theory and applications, Proc. Amer. Math. Soc. 110 (1990), no. 4, 961-969. https://doi.org/10.2307/2047743
  30. C. Perez Moreno, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), no. 1, 163-185. https://doi.org/10.1006/jfan.1995.1027
  31. C. Perez Moreno, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl. 3 (1997), no. 6, 743-756. https://doi.org/10.1007/BF02648265
  32. C. Perez Moreno and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. London Math. Soc. (2) 65 (2002), no. 3, 672-692. https://doi.org/10.1112/S0024610702003174
  33. M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, Inc., New York, 1991.
  34. C. Segovia Fernandez and J. Torrea, Vector-valued commutators and applications, Indiana Univ. Math. J. 38 (1989), no. 4, 959-971. https://doi.org/10.1512/iumj.1989.38.38044
  35. C. Segovia Fernandez and J. Torrea, Weighted inequalities for commutators of fractional and singular integrals, Publ. Mat. 35 (1991), no. 1, 209-235. https://doi.org/10.5565/PUBLMAT_35191_09
  36. C. P. Xie, Some estimates of commutators, Real Anal. Exchange 36 (2010/11), no. 2, 405-415. http://projecteuclid.org/euclid.rae/1321020508 1020508
  37. P. Zhang, Weighted estimates for maximal multilinear commutators, Math. Nachr. 279 (2006), no. 4, 445-462. https://doi.org/10.1002/mana.200310370
  38. P. Zhang, Multiple weighted estimates for commutators of multilinear maximal function, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 6, 973-994. https://doi.org/10.1007/s10114-015-4293-6
  39. P. Zhang, Z. Si, and J. L. Wu, Some notes on commutators of the fractional maximal function on variable Lebesgue spaces, J. Inequal. Appl. 2019 (2019), Paper No. 9, 17 pp. https://doi.org/10.1186/s13660-019-1960-7
  40. P. Zhang and J. L. Wu, Commutators of fractional maximal functions, Acta Math. Sinica (Chinese Ser.) 52 (2009), no. 6, 1235-1238.
  41. P. Zhang and J. L. Wu, Commutators for the maximal functions on Lebesgue spaces with variable exponent, Math. Inequal. Appl. 17 (2014), no. 4, 1375-1386. https://doi.org/10.7153/mia-17-101
  42. P. Zhang and J. L. Wu, Commutators of the fractional maximal function on variable exponent Lebesgue spaces, Czechoslovak Math. J. 64(139) (2014), no. 1, 183-197. https://doi.org/10.1007/s10587-014-0093-x