On commutators of Marcinkiewicz integrals with rough kernel

In this note we prove that the higher-order commutators mu(Omega,b)(m), mu(Omega,lambda,b)(*m) and mu(Omega,S,b)(m) are all of bounded operators on the weighted L-P spaces. These commutators are formed respectively by a BMO(R-n) function b(x) and a class of rough Marcinkiewicz integral mu(Omega), mu(Omega,lambda)(*) and mu(Omega,S), which are corresponding to the Littlewood-Paley g-function, Littlewood-Paley g(lambda)(*)-function and the Lusin area integral, respectively. The results in this paper are essential improvements and extensions of the results by Torchinsky and Wang (1990) and by Alvarez et al. (1993). (C) 2002 Elsevier Science (USA). All rights reserved.