$L^2$ boundedness for maximal commutators with rough variable kernels

Yanping Chen; Yong Ding; Ran Li

doi:10.4171/rmi/640

JournalsrmiVol. 27, No. 2pp. 361–391

$L^{2}$ 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

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Abstract

For $b \in BMO (R^{n})$ and $k \in N$ , the $k$ -th order maximal commutator of the singular integral operator $T$ with rough variable kernels is defined by

T_{b, k}^{*} f (x) = ε > 0 sup \int_{∣ x - y ∣ > ε} \frac{Ω ( x , x - y )}{∣ x - y ∣ ^{n}} (b (x) - b (y))^{k} f (y) d y .

In this paper the authors prove that the $k$ -th order maximal commutator $T_{b, k}^{*}$ is a bounded operator on $L^{2} (R^{n})$ if $Ω$ satisfies the same conditions given by Calderón and Zygmund. Moreover, the $L^{2}$ -boundedness of the $k$ -th order commutator of the rough maximal operator $M_{Ω}$ with variable kernel, which is defined by

M_{Ω; b, k} f (x) = r > 0 sup \frac{1}{r ^{n}} \int_{∣ x - y ∣ < r} ∣Ω (x, x - y) ∣∣ b (x) - b (y) ∣^{k} ∣ f (y) ∣ d y,

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, $L^{2}$ boundedness for maximal commutators with rough variable kernels. Rev. Mat. Iberoam. 27 (2011), no. 2, pp. 361–391

DOI 10.4171/RMI/640