WEIGHTED NORM INEQUALITIES FOR TOEPLITZ TYPE OPERATOR ASSOCIATED TO SINGULAR IN OPERATOR WITH NON SMOOTH KERNEL

Let T-k,T-1 be singular integrals with non-smooth kernels, which are associated with an approximation to the identity or +/- 1 (the identity operator), T-k,T-2 and T-k,T-4 are the linear operators, T-k,T-3 = +/- 1. Denote the Toeplitz type operator by T-b = Sigma(m)(k=1) (T-k,T-1 MbI alpha Tk,2 + (TI alpha MbTk,4)-I-k,3) where M-b f (x) = b(x)f(x), and 1(alpha) 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.