非齐型空间中一类满足Hrmander条件的参数型Marcinkiewicz积分交换子的估计

Estimates for Commutators of the Parameter Marcinkiewicz Integrals with Hrmander-type Condition in Non-homogeneous Spaces

设μ为R^d上的非负Radon测度,满足对固定的C_0>0和n∈(0,d],以及所有的x∈R^d和r>0,μ(B(x,r))≤C_0r^n.本文主要证明了由参数型Marcinkiewicz积分M~ρ和Lipschitz函数b生成的交换子M_b~ρ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M_b~ρ不仅从Lebesgue空间L^p(μ)到Lebesgue空间L^q(μ)有界,从Lebesgue空间L^p(μ)到Lipschitz空间Lip_(β-n/p)(μ)有界,且从Lipschitz空间Lip_(β-n/p)(μ)到空间RBMO(μ)有界.

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）.