On commutators of Marcinkiewicz integrals with rough kernel ✩

In this note we prove that the higher-order commutators μm Ω,b, μ∗,m Ω,λ,b and μm Ω,S,b are all of bounded operators on the weighted Lp spaces. These commutators are formed respectively by a BMO(Rn) function b(x) and a class of rough Marcinkiewicz integral operators μΩ, μ∗Ω,λ and μΩ,S, which are corresponding to the Littlewood–Paley gfunction, Littlewood–Paley g∗λ-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).  2002 Elsevier Science (USA). All rights reserved.