JournalsrmiVol. 27, No. 2pp. 361–391

L2L^2 boundedness for maximal commutators with rough variable kernels

  • Yanping Chen

    University of Sciences and Technology, Beijing, China
  • Yong Ding

    Beijing Normal University, China
  • Ran Li

    Beijing Normal University, China
$L^2$ boundedness for maximal commutators with rough variable kernels cover
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Abstract

For bBMO(Rn)b\in BMO(\mathbb{R}^n) and kNk\in\mathbb{N}, the kk-th order maximal commutator of the singular integral operator TT with rough variable kernels is defined by

Tb,kf(x)=supε>0xy>εΩ(x,xy)xyn(b(x)b(y))kf(y)dy.T^{\ast}_{b,k}f(x) = \sup_{\varepsilon > 0} \biggl| \int_{|x-y| > \varepsilon} \frac{\Omega(x,x-y)}{|x-y|^n} (b(x)-b(y))^{k} f(y) dy \biggl|.

In this paper the authors prove that the kk-th order maximal commutator Tb,kT^{\ast}_{b,k} is a bounded operator on L2(Rn)L^2(\mathbb{R}^n) if Ω\Omega satisfies the same conditions given by Calderón and Zygmund. Moreover, the L2L^2-boundedness of the kk-th order commutator of the rough maximal operator MΩM_\Omega with variable kernel, which is defined by

MΩ;b,kf(x)=supr>01rnxy<rΩ(x,xy)b(x)b(y)kf(y)dy,M_{\Omega;b,k}f(x) = \sup_{r > 0} \dfrac{1}{r^{n}} \int_{|x-y| < r} |\Omega(x,x-y)| |b(x)-b(y)|^{k} |f(y)| dy,

is also given here. These results obtained in this paper are substantial improvement and extension of some known results.

Cite this article

Yanping Chen, Yong Ding, Ran Li, L2L^2 boundedness for maximal commutators with rough variable kernels. Rev. Mat. Iberoam. 27 (2011), no. 2, pp. 361–391

DOI 10.4171/RMI/640