Tohoku Mathematical Journal

A revisit on commutators of linear and bilinear fractional integral operator

Mingming Cao and Qingying Xue

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $I_{\alpha}$ be the linear and $\mathcal{I}_{\alpha}$ be the bilinear fractional integral operators. In the linear setting, it is known that the two-weight inequality holds for the first order commutators of $I_{\alpha}$. But the method can't be used to obtain the two weighted norm inequality for the higher order commutators of $I_{\alpha}$. In this paper, using some known results, we first give an alternative simple proof for the first order commutators of $I_{\alpha}$. This new approach allows us to consider the higher order commutators. Then, by using the Cauchy integral theorem, we show that the two-weight inequality holds for the higher order commutators of $I_{\alpha}$. In the bilinear setting, we present a dyadic proof for the characterization between $BMO$ and the boundedness of $[b,\mathcal{I}_{\alpha}]$. Moreover, some bilinear paraproducts are also treated in order to obtain the boundedness of $[b,\mathcal{I}_{\alpha}]$.

Article information

Source
Tohoku Math. J. (2), Volume 71, Number 2 (2019), 303-318.

Dates
First available in Project Euclid: 21 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1561082600

Digital Object Identifier
doi:10.2748/tmj/1561082600

Mathematical Reviews number (MathSciNet)
MR3973253

Zentralblatt MATH identifier
07108041

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 47G10: Integral operators [See also 45P05]

Keywords
commutator paraproduct Haar function dyadic analysis bilinear fractional integral operators

Citation

Cao, Mingming; Xue, Qingying. A revisit on commutators of linear and bilinear fractional integral operator. Tohoku Math. J. (2) 71 (2019), no. 2, 303--318. doi:10.2748/tmj/1561082600. https://projecteuclid.org/euclid.tmj/1561082600


Export citation

References

  • S. Bloom, A commutator theorem and weighted BMO, Trans. Amer. Math. Soc. 292(1985), no. 1, 103–122.
  • L. Chaffee, Characterizations of bounded mean oscillation through commutators of bilinear singular integral operators, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 6, 1159–1166.
  • S. Chanillo, A note on commutators, Indiana Univ. Math. J. (1) 31 (1982), 7–16.
  • X. Chen and Q. Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl. 362 (2010), 355–373.
  • D. Cruz-Uribe and K. Moen, Sharp norm inequalities for commutators of classical operators, Publ. Mat. 56 (2011), no. 1, 147–190.
  • L. Dalenc and Y. Ou, Upper bound for multi-parameter iterated commutators, Publ. Mat. 60 (2016), 191–220.
  • S. H. Ferguson and M. T. Lacey, A characterization of product BMO by commutators, Acta Math. 189 (2002), no. 2, 143–160.
  • I. Holmes, R. Rahm and S. Spencer, Two-weight inequalities for commutators with fractional integral operators, Studia Math. 233 (2016), no. 3, 279–291.
  • I. Holmes, M. T. Lacey and B. D. Wick, Bloom's inequality: commutators in a two-weight setting, Arch. Math. 106 (2016), 53–63.
  • I. Holmes, M. T. Lacey and B. D. Wick, Bloom's inequality: commutators in a two-weight setting. (English summary), Arch. Math. (Basel) 106 (2016), no. 1, 53–63.
  • I. Holmes and B. D. Wick, Two weight inequalities for iterated commutators with Calderón-Zygmund operators, J. Operator Theory 79 (2018), no.1, 33–54.
  • T. P. Hytönen, On Petermichl's dyadic shift and the Hilbert transform, C. R. Math. Acad. Sci. Paris, 346 (2008), no. 21–22, 1133–1136.
  • T. P. Hytönen, The Holmes–Wick theorem on two-weight bounds for higher order commutators revisited, Arch. Math. 107 (2016), 389–395.
  • M. T. Lacey, Commutators with Reisz potentials in one and several parameters, Hokkaido Math. J. 36 (2007), no. 1, 175–191.
  • M. T. Lacey, S. Petermichl, J. C. Pipher and B. D. Wick, Multiparameter Riesz Commutators, Amer. J. Math. 131 (2009), no. 3, 731–769.
  • A. K. Lerner, S. Ombrosi and I. P. Rivera-Ríos, On pointwise and weighted estimates for commutators of Calderón-Zygmund operators, Adv. Math. 319 (2017), 153–181.
  • S. Petermichl, S. Treil and A. Volberg, Why the Riesz transforms are averages of the dyadic shifts?, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations(El Escorial, 2000), 2002.