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

For b ∈ BMO(Rn) and k ∈ N, the k-th order maximal commutator of the singular integral operator T with rough variable kernels is defined by T ∗ b,kf(x) = sup ε>0 ∣∣∣|x−y|>ε Ω(x, x − y) |x − y|n (b(x) − b(y))f(y)dy ∣∣∣∣. In this paper the authors prove that the k-th order maximal commutator T ∗ b,k is a bounded operator on L 2(Rn) if Ω satisfies the same conditions given by Calderón and Zygmund. Moreover, the L2-boundedness of the k-th order commutator of the rough maximal operator MΩ with variable kernel, which is defined by MΩ;b,kf(x) = sup r>0 1 rn ∫ |x−y|

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