中文摘要：

设μ为Rd上的非负Radon测度,满足对固定的C0〉0和n∈（0,d],以及所有的x∈Rd和r〉0,μ（B（x,r））≤C0rn.本文主要证明了由参数型Marcinkiewicz积分Mρ和Lipschitz函数b生成的交换子Mbρ的有界性.在M的核函数满足较强的Hrmander条件下,作者证明了Mbρ不仅从Lebesgue空间Lp（μ）到Lebesgue空间Lq（μ）有界,从Lebesgue空间Lp（μ）到Lipschitz空间Lipβ-n/p（μ）有界,且从Lipschitz空间Lipβ-n/p（μ）到空间RBMO（μ）有界.

英文摘要：

Let μ be a nonnegative Radon on Rd,and μ satisfy the condition μ（B（x,r））≤C0rn for any x∈Rd,r〉0 and some fixed n∈（0,d].In this paper,the authors prove the boundedness of the commutator Mbρ generated by the parameter Marcinkiewicz integral Mρwith Lipschitz function b.Under the assumption that the kernel of M satisfies certain slightly stronger Hrmander-type condition,the authors prove that Mρbis not only bounded from the Lebesgue space Lp（μ）to the Lebesgue space Lq（μ）,and from the Lebesgue space Lp（μ） to the Lipschitz space Lipβ-n/p（μ）,but also Mbρ is bounded from the Lipschitz space Lipβ-n/p（μ）to the space RBMO（μ）.