Taiwanese Journal of Mathematics


Suzhen Mao, Yoshihiro Sawano, and Huoxiong Wu

Full-text: Open access


Let $b \in \operatorname{BMO}(\mathbb{R}^n)$ and $\mathscr{M}_\Omega$ be the Marcinkiewicz integral operatorwith kernel $\frac{\Omega(x)}{|x|^{n-1}}$, where $\Omega$ is homogeneous of degree zero, integrable and has mean value zero on the unit sphere $S^{n-1}$. In this paper, by means of Fourier transform estimates and approximationto the operator $\mathscr{M}_\Omega$ with integral operators having smooth kernelswe show that if $b \in \operatorname{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain weak size condition, then the commutator $\mathscr{M}_{\Omega,b} = [b, \mathscr{M}_\Omega]$ generated by $b$ and $\mathscr{M}_\Omega$ is a compact operator on $L^p(\mathbb{R}^n)$ for some $1\lt p\lt \infty$.

Article information

Taiwanese J. Math., Volume 19, Number 6 (2015), 1777-1793.

First available in Project Euclid: 4 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 11L15: Weyl sums 11P05: Waring's problem and variants

commutators Marcinkiewicz integral operators compactness approximation


Mao, Suzhen; Sawano, Yoshihiro; Wu, Huoxiong. ON THE COMPACTNESS OF COMMUTATORS FOR ROUGH MARCINKIEWICZ INTEGRAL OPERATORS. Taiwanese J. Math. 19 (2015), no. 6, 1777--1793. doi:10.11650/tjm.19.2015.5656. https://projecteuclid.org/euclid.twjm/1499133739

Export citation


  • H. Al-Qassem, A. Al-Salman, L. C. Cheng and Y. Pan, $L^p$ bounds for the function of Marcinkiewicz, Math. Res. Lett. 9 (2002), 697–700.
  • M. Berger, Nonlinearity and Functional Analysis, New York, Academic Press, 1962, 64–107.
  • A. Benedek, A. Calderón and R. Panzone, Convolution operators on Banach space value functions, Proc. Nat. Acd. Sci. USA 48 (1962), 356–365.
  • G. Bourdaud, M. Lanze de Cristoforis and W. Sickel, Functional calculus on BMO and related spaces, J. Funct. Anal. 189 (2002), 515–538.
  • J. Chen, D. Fan and Y. Pan, A note on a Marcinkiewicz integral operator, Math. Nachr. 227 (2001), 33–42.\! $<$33::aid-mana33$>$3.0.co;2-0
  • J. Chen and G. Hu, Compact commutators of rough singular integral operators, Canad. Math. Bull. 58 (2015), 19–29.
  • Y. Chen and Y. Ding, Compactness characterization of commutators for Littlewood-Paley operators, Kodai Math. J. 32 (2009), 256–323.
  • Y. Chen, Y. Ding and X. Wang, Compactness for commutators of Marcinkiewicz integrals in Morrey spaces, Taiwanese J. Math. 15(2) (2011), 633–658.
  • Y. Chen, K. Zhu and Y. Ding, On the compactness of the commutator of parabolic Marcinkiewicz integral with variable kernel, J. Function Spaces 2014 (2014), Article ID 693534, 12 pages.
  • R. Coifman and G. Weiss, Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645.
  • Y. Ding, D. Fan and Y. Pan, $L^p$-boundedness of Marcinkiewicz integrals with Hardy space function kernel, Acta Math. Sinica 16 (2000), 593–600.
  • Y. Ding, S. Lu and K. Yabuta, A problem on rough Marcinkiewicz functions, J. Austral. Math. Soc. 71 (2001), 1–9.
  • ––––, On commutators of Marcinkiewicz integrals with rough kernel, J. Math. Anal. Appl. 275 (2002), 60–68.
  • Y. Ding, T. Mei and Q. Xue, Compactness of maximal commutators of bilinear Calderon-Zygmund singular integral operators, arXiv: 1310.5787v3, 13 Dec. 2013.
  • L. Grafakos and A. Stefanov, $L^p$ bounds for singular integrals and maximal singular integrals with rough kernels, Indiana Univ. Math. J. 47 (1998), 455–469.
  • G. Hu, $L^p(\mathbb R^n)$ boundedness for a class of $g$-functions and applications, J. Hokkaido Math. 32 (2003), 497–521.
  • G. Hu and D. Yan, On commutator of the Marcinkiewicz integral, J. Math. Anal. Appl. 283 (2003), 351–361.
  • E. M. Stein, On the function of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430–466.
  • A. Torchinsky and S. Wang, A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), 235–243.
  • A. Uchiyama, On the compactness of operators of Hankel type, J. Tohoku Math. 30 (1978), 163–171.
  • T. Walsh, On the function of Marcinkiewicz, Studia Math. 44 (1972), 203–217.
  • H. Wu, On Marcinkiewicz integral operators with rough kernels, Integral Equations Operator Theory 52 (2005), 285–298.
  • ––––, $L^p$ bounds for Marcinkiewicz integrals associated to surfaces of revolution, J. Math. Anal. Appl. 321(2) (2006), 811–827.
  • ––––, $L^p$ estimates for the commutators of Marcinkiewicz integrals with kernels belonging to certain block spaces, Math. Nachr. 279(9-10) (2006), 1128–1144.
  • K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1995.