Weighted Boundedness of Commutators Related to Marcinkiewicz Integrals

In this paper,applying the atomic decomposition of the real weighted Hardyspaces H_ω~p(R~n),the authors study the boundedness of the operatorμ_Ω~b,the commutator generatedby a function b∈A_β(R~n)(0 β≤1) and the Marcinkiewicz integralsμ_Ω.It is shownthatμ_Ω~b is bounded from L~p(ω~p) to L~q(ω~q),from L~1(|x|(γ(n-β))/n) to weak Ln/(n-β)(|x|~γ),and fromH~p(ω~p) to L~q(ω~q) in the case that 1/n-1/q=β/n.