ON THE COMPACTNESS OF COMMUTATORS FOR ROUGH MARCINKIEWICZ INTEGRAL OPERATORS

Let b is an element of BMO(R-n) and M-Omega be the Marcinkiewicz integral operator with kernel Omega(x)/vertical bar x vertical bar(n-1), where Omega is homogeneous of degree zero, integrable and has mean value zero on the unit sphere Sn-1. In this paper, by means of Fourier transform estimates and approximation to the operator M-Omega with integral operators having smooth kernels we show that if b is an element of CMO(R-n) and (Omega) satisfies a certain weak size condition, then the commutator M-Omega,M-b = [b, M-Omega] generated by b and M-Omega is a compact operator on L-p(R-n) for some 1 < p < infinity.