Operators with rough singular kernels

For α >0, we study the singular integral operators TΩ,α and the Marcinkiewicz integral operator μΩ,α. The kernels of these operators behave like |y|−n−α near y = 0, and contain a distribution Ω on the unit sphere Sn−1. We prove that if Ω ∈ Hr (Sn−1) (r = (n − 1)/(n − 1 + α)) satisfying certain cancellation condition, then both TΩ,α and μΩ,α can be extend to be the bounded operators from the Sobolev space L pα (Rn) to the Lebesgue space Lp(Rn). The result improves and extends some known results. © 2007 Elsevier Inc. All rights reserved