Weighted norm inequalities for Toeplitz type operator associated to singular integral operator with non-smooth kernel

Let T k,1 be singular integrals with non-smooth kernels, which are associated with an approximation to the identity or ±I (the identity operator), T k,2 and T k,4 are the linear operators, T k,3 = ±I .Denote the Toeplitz type operator bywhere M b f (x) = b(x) f (x), and I α is the fractional integral operator.In this paper, we establish the sharp maximal function estimates for T b when b belongs to weighted Lipschitz function space.As an application, the boundedness of the operator on weighted Lebesgue space is obtained.