Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON WEIGHTED COMPACTNESS OF COMMUTATORS OF BILINEAR FRACTIONAL MAXIMAL OPERATOR

  • He, Qianjun (School of Applied Science Beijing Information Science and Technology University) ;
  • Zhang, Juan (School of Science Beijing Forestry University)
  • Received : 2021.03.17
  • Accepted : 2021.08.19
  • Published : 2022.05.01
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Abstract

Let Mα be a bilinear fractional maximal operator and BMα be a fractional maximal operator associated with the bilinear Hilbert transform. In this paper, the compactness on weighted Lebesgue spaces are considered for commutators of bilinear fractional maximal operators; these commutators include the fractional maximal linear commutators Mjα,β and BMjα,β (j = 1, 2), the fractional maximal iterated commutator ${\mathcal{M}}_{{\alpha},{\vec{b}}}$, and $BM_{{\alpha},{\vec{b}}}$, where b ∈ BMO(ℝd) and ${\vec{b}}\;=\;(b_1,b_2)\;{\in}\;BMO({\mathbb{R}}^d)\;{\times}\;BMO({\mathbb{R}}^d)$. In particular, we improve the well-known results to a larger scale for 1/2 < q < ∞ and give positive answers to the questions in [2].

Keywords

Acknowledgement

The first author was in part supported by National Natural Science Foundation of China (Nos. 11871452, 12071473) and the second author was supported by National Natural Science Foundation of China (No. 12101049).

References

  1. F. Beatrous and S.-Y. Li, On the boundedness and compactness of operators of Hankel type, J. Funct. Anal. 111 (1993), no. 2, 350-379. https://doi.org/10.1006/jfan.1993.1017
  2. A. Benyi, W. Damian, K. Moen, and R. H. Torres, Compactness properties of commutators of bilinear fractional integrals, Math. Z. 280 (2015), no. 1-2, 569-582. https://doi.org/10.1007/s00209-015-1437-4
  3. A. Benyi, W. Damian, K. Moen, and R. H. Torres, Compact bilinear commutators: the weighted case, Michigan Math. J. 64 (2015), no. 1, 39-51. https://doi.org/10.1307/mmj/1427203284
  4. A. Benyi and R. H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3609-3621. https://doi.org/10.1090/S0002-9939-2013-11689-8
  5. R. Bu and J. Chen, Compactness for the commutators of multilinear singular integral operators with non-smooth kernels, Appl. Math. J. Chinese Univ. Ser. B 34 (2019), no. 1, 55-75. https://doi.org/10.1007/s11766-019-3501-z
  6. M. Cao, A. Olivo, and K. Yabuta, Extrapolation for multilinear compact operators and applications, Preprint, 2020, arXiv:2011.13191.
  7. L. Chaffee and R. H. Torres, Characterization of compactness of the commutators of bilinear fractional integral operators, Potential Anal. 43 (2015), no. 3, 481-494. https://doi.org/10.1007/s11118-015-9481-6
  8. Y. Chen and Y. Ding, Compactness of commutators of singular integrals with variable kernels, Chinese J. Contemp. Math. 30 (2009), no. 2, 153-166; translated from Chinese Ann. Math. Ser. A 30 (2009), no. 2, 201-212.
  9. Y. Chen and Y. Ding, Compactness characterization of commutators for Littlewood-Paley operators, Kodai Math. J. 32 (2009), no. 2, 256-323. http://projecteuclid.org/euclid.kmj/1245982907 https://doi.org/10.2996/kmj/1245982907
  10. X. Chen and Q. Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl. 362 (2010), no. 2, 355-373. https://doi.org/10.1016/j.jmaa.2009.08.022
  11. R. R. Coifman and C. 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
  12. J. B. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics, 96, Springer-Verlag, New York, 1985. https://doi.org/10.1007/978-1-4757-3828-5
  13. W. Guo, Y. Wen, H. Wu, and D. Yang, On the compactness of oscillation and variation of commutators, Banach J. Math. Anal. 15 (2021), no. 2, Paper No. 37, 29 pp. https://doi.org/10.1007/s43037-021-00123-z
  14. Q. He and P. Li, On weighted compactness of commutators of Schrodinger operators, Preprint, 2021, arXiv: 2102.01277.
  15. Q. He, M. Wei, and D. Yan, Weighted estimates for bilinear fractional integral operators and their commutators on Morrey spaces, Preprint, 2019, arXiv:1905.10946.
  16. Q. He and D. Yan, Bilinear fractional integral operators on Morrey spaces, Positivity 25 (2021), no. 2, 399-429. https://doi.org/10.1007/s11117-020-00763-9
  17. T. Iida, A characterization of a multiple weights class, Tokyo J. Math. 35 (2012), no. 2, 375-383. https://doi.org/10.3836/tjm/1358951326
  18. T. Iida, Weighted estimates of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces, J. Inequal. Appl. 2016 (2016), Paper No. 4, 23 pp. https://doi.org/10.1186/s13660-015-0953-4
  19. T. Iwaniec and C. Sbordone, Riesz transforms and elliptic PDEs with VMO coefficients, J. Anal. Math. 74 (1998), 183-212. https://doi.org/10.1007/BF02819450
  20. S. G. Krantz and S.-Y. Li, Boundedness and compactness of integral operators on spaces of homogeneous type and applications. I, J. Math. Anal. Appl. 258 (2001), no. 2, 629-641. https://doi.org/10.1006/jmaa.2000.7402
  21. S. S. Kutateladze, Fundamentals of Functional Analysis, Kluwer Texts in the Mathematical Sciences, 12, Kluwer Academic Publishers Group, Dordrecht, 1996. https://doi.org/10.1007/978-94-015-8755-6
  22. A. K. Lerner, S. Ombrosi, C. Perez, R. H. Torres, and R. Trujillo-Gonzalez, New maximal functions and multiple weights for the multilinear Calderon-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222-1264. https://doi.org/10.1016/j.aim.2008.10.014
  23. X. Li, Q. He, and D. Yan, Weighted estimates for bilinear fractional integral operator of iterated product commutators on Morrey spaces, J. Math. Inequal. 14 (2020), no. 4, 1249-1267. https://doi.org/10.7153/jmi-2020-14-81
  24. P. Li and L. Peng, Compact commutators of Riesz transforms associated to Schrodinger operator, Pure Appl. Math. Q. 8 (2012), no. 3, 713-739. https://doi.org/10.4310/PAMQ.2012.v8.n3.a7
  25. H. Liu and L. Tang, Compactness for higher order commutators of oscillatory singular integral operators, Internat. J. Math. 20 (2009), no. 9, 1137-1146. https://doi.org/10.1142/S0129167X09005698
  26. K. Moen, Weighted inequalities for multilinear fractional integral operators, Collect. Math. 60 (2009), no. 2, 213-238. https://doi.org/10.1007/BF03191210
  27. B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274. https://doi.org/10.2307/1996833
  28. 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.
  29. A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Math. J. (2) 30 (1978), no. 1, 163-171. https://doi.org/10.2748/tmj/1178230105
  30. S. Wang and Q. Xue, On weighted compactness of commutators of bilinear maximal Calderon-Zygmund singular integral operators, Forum Math. 34 (2022), no. 2, 307-322. https://doi.org/10.1515/forum-2020-0357
  31. D.-H. Wang, J. Zhou, and Z.-D. Teng, On the compactness of commutators of Hardy-Littlewood maximal operator, Anal. Math. 45 (2019), no. 3, 599-619. https://doi.org/10.1007/s10476-019-0818-z
  32. Q. Xue, Weighted estimates for the iterated commutators of multilinear maximal and fractional type operators, Studia Math. 217 (2013), no. 2, 97-122. https://doi.org/10.4064/sm217-2-1
  33. Q. Xue, K. Yabuta, and J. Yan, Weighted Frechet-Kolmogorov theorem and compactness of vector-valued multilinear operators, J. Geom. Anal. 31 (2021), no. 10, 9891-9914. https://doi.org/10.1007/s12220-021-00630-3