In this paper the author proves that the commutator of the Marcinkiewicz integral operator with rough variable kernel is bounded from the homogeneous Sobolev space Lγ^2（R^n） to the Lebesgue space L^2（R^n）, which is a substantial improvement and extension of some known results.
Let b ∈ L loc(? n ) and L denote the Littlewood-Paley operators including the Littlewood-Paley g function, Lusin area integral and g λ * function. In this paper, the authors prove that the L p boundedness of commutators [b, L] implies that b ∈ BMO(? n ). The authors therefore get a characterization of the L p -boundedness of the commutators [b, L]. Notice that the condition of kernel function of L is weaker than the Lipshitz condition and the Littlewood-Paley operators L is only sublinear, so the results obtained in the present paper are essential improvement and extension of Uchiyama’s famous result.
