Weighted Norm Inequalities for Toeplitz Type Operator Related to Singular Integral Operator with Variable Kernel

Abstract

Let $T^{k,1}$ be the singular integrals with variable Calderόn-Zygmund kernels or $\pm I$ (the identity operator), let $T^{k,2}$ and $T^{k,4}$ be the linear operators, and let $T^{k,3}=\pm I$. Denote the Toeplitz type operator by

$$T^b=\sum_{k=1}^t(T^{k,1}M^bI_\alpha T^{k,2}+T^{k,3}I_\alpha M^b T^{k,4}),$$

where $M^bf=bf,$ and $I_\alpha$ is the fractional integral operator. In this paper, we investigate the boundedness of the operator on weighted Lebesgue space when $b$ belongs to weighted Lipschitz space.