Compact Commutators of Rough Singular Integral Operators

Let b is an element of BMO(R-n) and T-Omega be the singular integral operator with kernel Omega(x)/vertical bar x vertical bar(n), where Omega is homogeneous of degree zero, integrable, and has mean value zero on the unit sphere Sn-1. In this paper, using Fourier transform estimates and approximation to the operator T-Omega by integral operators with smooth kernels, it is proved that if b is an element of CMO(R-n) and Omega satisfies certain minimal size condition, then the commutator generated by b and T-Omega is a compact operator on L-p (R-n) for appropriate index p. The associated maximal operator is also considered.