Toeplitz Type Operator Associated to Singular Integral Operator with Variable Kernel on Weighted Morrey Spaces

Abstract

Suppose $T^{k,1}$ and $T^{k,2}$ are singular integrals with variable kernels and mixed homogeneity or $\pm I$ (the identity operator). Denote the Toeplitz type operator by\begin{align*}T^b=\sum_{k=1}^QT^{k,1}M^bT^{k,2}, \end{align*} where $M^bf=bf.$ In this paper, the boundedness of $T^b$ on weighted Morrey space are obtained when $b$ belongs to the weighted Lipschitz function space and weighted BMO function space, respectively.