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 $$\int_0^1t^{-(\alpha+n/q_2-\lambda)}\psi(t)\log{\frac{2}{t}}dt\infty$$ . It turned to be $\int_0^1t^{-n/p}\psi(t)\log{\frac{2}{t}}dt\infty$ as $\alpha=0$, $\lambda=0$ and $q_1=q_2=p1$, and then $U_{\psi}^b$ is bounded on $L^p(R^n)$.