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XMO and Weighted Compact Bilinear Commutators

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Abstract

To study the compactness of bilinear commutators of certain bilinear Calderón–Zygmund operators which include (inhomogeneous) Coifman–Meyer bilinear Fourier multipliers and bilinear pseudodifferential operators as special examples, Torres and Xue (Rev Mat Iberoam 36:939–956, 2020) introduced a new subspace of BMO\(\,(\mathbb {R}^n)\), denoted by XMO\(\,(\mathbb {R}^n)\), and conjectured that it is just the space VMO\(\,(\mathbb {R}^n)\) introduced by D. Sarason. In this article, the authors give a negative answer to this conjecture by establishing an equivalent characterization of XMO\(\,(\mathbb {R}^n)\), which further clarifies that XMO\(\,(\mathbb {R}^n)\) is a proper subspace of VMO\(\,(\mathbb {R}^n)\). This equivalent characterization of XMO\(\,(\mathbb {R}^n)\) is formally similar to the corresponding one of CMO\(\,(\mathbb {R}^n)\) obtained by A. Uchiyama, but its proof needs some essential new techniques on dyadic cubes as well as some exquisite geometrical observations. As an application, the authors also obtain a weighted compactness result on such bilinear commutators, which optimizes the corresponding result in the unweighted setting.

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References

  1. Bényi, A., Torres, R.H.: Symbolic calculus and the transposes of bilinear pseudodifferential operators. Commun. Partial Differ. Equ. 28, 1161–1181 (2003)

    Article  MathSciNet  Google Scholar 

  2. Bényi, A., Torres, R.H.: Compact bilinear operators and commutators. Proc. Am. Math. Soc. 141, 3609–3621 (2013)

    Article  MathSciNet  Google Scholar 

  3. Bényi, A., Damián, W., Moen, K., Torres, R.H.: Compact bilinear commutators: the weighted case. Mich. Math. J. 64, 39–51 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bényi, A., Martell, J.M., Moen, K., Stachura, E., Torres, R.H.: Boundedness results for commutators with BMO functions via weighted estimates: a comprehensive approach. Math. Ann. 376, 61–102 (2020)

    Article  MathSciNet  Google Scholar 

  5. Bui, T.A., Duong, X.T.: Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers. Bull. Sci. Math. 137, 63–75 (2013)

    Article  MathSciNet  Google Scholar 

  6. Cao, M., Olivo, A., Yabuta, K.: Extrapolation for multilinear compact operators and applications, arXiv:2011.13191

  7. Cao, M., Olivo, A., Yabuta, K.: Weighted interpolation for multilinear compact operators, Preprint

  8. Chaffee, L., Chen, P., Han, Y., Torres, R.H., Ward, L.A.: Characterization of compactness of commutators of bilinear singular integral operators. Proc. Am. Math. Soc. 146, 3943–3953 (2018)

    Article  MathSciNet  Google Scholar 

  9. Chen, P., Duong, X.T., Li, J., Wu, Q.: Compactness of Riesz transform commutator on stratified Lie groups. J. Funct. Anal. 277, 1639–1676 (2019)

    Article  MathSciNet  Google Scholar 

  10. Clop, A., Cruz, V.: Weighted estimates for Beltrami equations. Ann. Acad. Sci. Fenn. Math. 38, 91–113 (2013)

    Article  MathSciNet  Google Scholar 

  11. Coifman, R.R., Meyer, Y.: Commutateurs d’ intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28, 177–202 (1978)

    Article  MathSciNet  Google Scholar 

  12. Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103, 611–635 (1976)

  13. Cordes, H.O.: On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. J. Funct. Anal. 18, 115–131 (1975)

    Article  MathSciNet  Google Scholar 

  14. Ding, Y., Mei, T., Xue, Q.: Compactness of maximal commutators of bilinear Calderón–Zygmund singular integral operators. In: Some Topics in Harmonic Analysis and Applications, pp. 163–175, Adv. Lect. Math. (ALM), 34, Int. Press, Somerville, MA, (2016)

  15. Grafakos, L., Torres, R.H.: Multilinear Calderón–Zygmund theory. Adv. Math. 165, 124–164 (2002)

  16. Grafakos, L., Van Nguyen, H.: Multilinear Fourier multipliers with minimal Sobolev regularity, I. Colloq. Math. 144, 1–30 (2016)

    Article  MathSciNet  Google Scholar 

  17. Grafakos, L., Miyachi, A., Van Nguyen, H., Tomita, N.: Multilinear Fourier multipliers with minimal Sobolev regularity, II. J. Math. Soc. Jpn. 69, 529–562 (2017)

    Article  MathSciNet  Google Scholar 

  18. He, Z., Han, Y., Li, J., Liu, L., Yang, D., Yuan, W.: A complete real-variable theory of Hardy spaces on spaces of homogeneous type. J. Fourier Anal. Appl. 25, 2197–2267 (2019)

    Article  MathSciNet  Google Scholar 

  19. Hu, G.: Compactness of the commutator of bilinear Fourier multiplier operator. Taiwanese J. Math. 18, 661–675 (2014)

    Article  MathSciNet  Google Scholar 

  20. Hytönen, T.: Extrapolation of compactness on weighted spaces, arXiv:2003.01606

  21. Hytönen, T., Lappas, S.: Extrapolation of compactness on weighted spaces II: off-diagonal and limited range estimates, arXiv:2006.15858

  22. Hytönen, T., Lappas, S.: Extrapolation of compactness on weighted spaces III: bilinear operators, arXiv:2012.10407

  23. Janson, S.: Mean oscillation and commutators of singular integral operators. Ark. Mat. 16, 263–270 (1978)

    Article  MathSciNet  Google Scholar 

  24. Lerner, A., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 220, 1222–1264 (2009)

    Article  MathSciNet  Google Scholar 

  25. Li, J., Nguyen, T.T., Ward, L.A., Wick, B.D.: The Cauchy integral, bounded and compact commutators. Studia Math. 250, 193–216 (2020)

    Article  MathSciNet  Google Scholar 

  26. Li, K.: Multilinear commutators in the two-weight setting, arXiv:2006.09071

  27. Pérez, C., Torres, R. H.: Sharp maximal function estimates for multilinear singular integrals. In: Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001), pp. 323–331, Contemp. Math. 320, Amer. Math. Soc. Providence, RI (2003)

  28. Sarason, D.: Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207, 391–405 (1975)

    Article  MathSciNet  Google Scholar 

  29. Tang, L.: Weighted estimates for vector-valued commutators of multilinear operators. Proc. R. Soc. Edinb. Sect. A 138, 897–922 (2008)

    Article  MathSciNet  Google Scholar 

  30. Tao, J., Yang, Da, Yang, Do: Beurling–Ahlfors commutators on weighted Morrey spaces and applications to Beltrami equations. Potential Anal. 53, 1467–1491 (2020)

    Article  MathSciNet  Google Scholar 

  31. Torres, R.H., Xue, Q.: On compactness of commutators of multiplication and bilinear pesudodifferential operators and a new subspace of BMO. Rev. Mat. Iberoam. 36, 939–956 (2020)

    Article  MathSciNet  Google Scholar 

  32. Torres, R.H., Xue, Q., Yan, J.: Compact bilinear commutators: the quasi-Banach space case. J. Anal. 26, 227–234 (2018)

    Article  MathSciNet  Google Scholar 

  33. Uchiyama, A.: On the compactness of operators of Hankel type. Tôhoku Math. J. (2) 30, 163–171 (1978)

  34. Xue, Q., Yabuta, K., Yan, J.: Weighted Fréchet–Kolmogorov theorem and compactness of vector-valued multilinear operators. J. Geom. Anal. (2021). https://doi.org/10.1007/s12220-021-00630-3

  35. Yosida, K.: Functional Analysis, Reprint of the sixth (1980) edition, Classics in Mathematics. Springer, Berlin (1995)

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Acknowledgements

The authors Jin Tao and Dachun Yang would like to express their sincerely thanks to Professor Loukas Grafakos for his very helpful discussion on the subject of this article.

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Correspondence to Dachun Yang.

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Communicated by Loukas Grafakos.

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This project is partially supported by the National Natural Science Foundation of China (Grant Nos. 11971058, 12071197, 11871101 and 11871100).

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Tao, J., Xue, Q., Yang, D. et al. XMO and Weighted Compact Bilinear Commutators. J Fourier Anal Appl 27, 60 (2021). https://doi.org/10.1007/s00041-021-09854-x

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