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Compactness properties of commutators of bilinear fractional integrals


Commutators of a large class of bilinear operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be jointly compact. Under a similar commutation, fractional integral versions of the bilinear Hilbert transform yield separately compact operators.

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  1. The notion \(CMO\) for this space is not uniformly used throughout the literature. See [1] for remarks and references about this notation.

  2. We actually obtain estimates for these terms that slightly improve on the corresponding ones for \({\upalpha }=0\) in [1].


  1. Bényi, Á., Torres, R.H.: Compact bilinear operators and commutators. Proc. Am. Math. Soc. 141(10), 3609–3621 (2013)

    Article  MATH  Google Scholar 

  2. Bernicot, F., Maldonado, D., Moen, K., Naibo, V.: Bilinear Sobolev–Poincaré inequalities and Leibniz-type rules. J. Geom. Anal. 24, 1144–1180 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  3. Betancor, J.J., Fariña, J.C.: A note on compactness of commutators for fractional integrals associated with nondoubling measures. Z. Anal. Anwend. 26, 331–339 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Studia Math. 24, 113–190 (1964)

    MATH  MathSciNet  Google Scholar 

  5. Chanillo, S.: A note on commutators. Indiana Univ. Math. J. 31, 7–16 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, W., Sawyer, E.: A note on commutators of fractional integrals with \(RBMO(\mu )\) functions. Illinois J. Math. 46, 1287–1298 (2002)

    MATH  MathSciNet  Google Scholar 

  7. Chen, X., Xue, Q.: Weighted estimates for a class of multilinear fractional type operators. J. Math. Anal. Appl. 362, 355–373 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, Y., Ding, Y., Wang, X.: Compactness of commutators of Riesz potential on Morrey spaces. Potential Anal. 30, 301–313 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Coifman, R.R., Lions, P.L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Grafakos, L.: On multilinear fractional integrals. Studia Math. 102, 49–56 (1992)

    MATH  MathSciNet  Google Scholar 

  12. Grafakos, L., Kalton, N.: Some remarks on multilinear maps and interpolation. Math. Ann. 319, 151–180 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Iwaniec, T.: Nonlinear commutators and Jacobians. J. Fourier Anal. Appl. 3, 775–796 (2007)

    Article  MathSciNet  Google Scholar 

  15. Iwaniec, T., Sbordone, C.: Riesz transform and elliptic PDEs with VMO coefficients. J. Anal. Math. 74, 183–212 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kenig, C., Stein, E.: Multilinear estimates and fractional integration. Math. Res. Lett. 6, 1–15 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. 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  MATH  MathSciNet  Google Scholar 

  18. Lian, J., Wu, H.: A class of commutators for multilinear fractional integrals in nonhomogeneous spaces. J. Inequal. Appl., vol. 2008, Article ID 373050, 17 pages

  19. Moen, K.: New weighted estimates for bilinear fractional integrals. Trans. Am. Math. Soc. 366, 627–646 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  20. Pérez, C., Pradolini, G., Torres, R.H., Trujillo-González, R.: End-points estimates for iterated commutators of multilinear singular integrals. Bull. Lond. Math. Soc. 46(1), 26–42 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Pérez, C., Torres, R.H.: Sharp maximal function estimates for multilinear singular integrals. Contemp. Math. 320, 323–331 (2003)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  23. Uchiyama, A.: On the compactness of operators of Hankel type. Tôhoku Math. J. 30(1), 163–171 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  24. Wang, S.: The compactness of the commutator of fractional integral operator (in Chinese). Chin. Ann. Math. 8(A), 475–482 (1987)

    MATH  Google Scholar 

  25. Yosida, K.: Functional Analysis. Springer, Berlin (1995)

    Google Scholar 

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This work was positively impacted by the interactions that occurred during Bényi’s and Torres’ stay at the Erwin Schrödinger Institute (ESI), Vienna, Austria, for the special semester on Modern Methods of Time-Frequency Analysis II. They wish to express their gratitude to the ESI and the organizers of the event for their support and warm hospitality.

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Correspondence to Árpád Bényi.

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Á. B. partially supported by a grant from the Simons Foundation (No. 246024). K. M. and R. H. T. partially supported by NSF Grants 1201504 and 1069015, respectively.

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Bényi, Á., Damián, W., Moen, K. et al. Compactness properties of commutators of bilinear fractional integrals. Math. Z. 280, 569–582 (2015).

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