The Boundedness of Commutator Associated to Weighted Hardy-Littlewood Average Operator in Herz-type Spaces

The boundedness of commutator U_\psi^b generated by the Weighted Hardy-Littlewood average operator U_\phi and BMO function b in Herz and Morrey-Herz type spaces are discussed. It is showed that the sufficient condition for its boundedness in the Morrey-Herz type spaces is \int_0^1t^-(\alpha+n/q_2-\lambda)\psi(t)\log\frac2tdt\infty. It turned to be \int_0^1t^-n/p\psi(t)\log\frac2tdt\infty as \alpha=0, \lambda=0 and q_1=q_2=p1, and then U_\psi^b is bounded on L^p(R^n).