Compactness for the commutators of singular integral operators with rough variable kernels

Let TΩTΩ be the singular integral operator with variable kernel defined byTΩf(x)=p. v.∫RnΩ(x,x−y)|x−y|nf(y)dy, where Ω(x,y) is homogeneous of degree zero in the second variable y  , and ∫Sn−1Ω(x,z′)dσ(z′)=0 for any x∈Rnx∈Rn. In this paper, the authors prove that if Ω∈L∞(Rn)×Lq(Sn−1)Ω∈L∞(Rn)×Lq(Sn−1) for some q>2(n−1)/nq>2(n−1)/n, then the commutator generated by a CMO(Rn)CMO(Rn) function and TΩTΩ, and the associated lacunary maximal operator, are compact on L2(Rn)L2(Rn). The associated continuous maximal commutator is also considered.