An Endpoint Estimate for the Commutators of Singular Integral Operators with Rough Kernels

Let Omega be homogeneous of degree zero and have mean value zero on the unit sphere Sd-1, T-Omega be the homogeneous singular integral operator with kernel Omega(x)/vertical bar x vertical bar(d) and T-Omega,T-b be the commutator of T-Omega with symbol b. In this paper, we prove that if Omega is an element of L(log L)(2)(Sd-1), then for b is an element of BMO(R-d), T-Omega,T-b satisfies an endpoint estimate of L log L type.