Compactness of the commutators of homogeneous singular integral operators

Let TΩ be the singular integral operator with kernel Ω(x)/|x|~n,where Ω is homogeneous of degree zero,integrable and has mean value zero on the unit sphere S.In this paper,by Fourier transform estimates,Littlewood-Paley theory and approximation,the authors prove that if Ω∈L(lnL)~2(S),then the commutator generated by Tand CMO(R~n) function,and the corresponding discrete maximal operator,are compact on L~p(R~n,|x|) for p∈(1,∞) and γ∈(-1,p-1).