The paper introduces singular integral operators of a new type defined in the space L ^p with the weight function on the complex plane. For these operators, norm estimates are derived. Namely, if V is a complex-valued function on the complex plane satisfying the condition {pipe}V(z) - V(ζ){pipe} ≤ w{pipe}z - ζ{pipe} and F is an entire function, then we put, It is shown that if the weight function ω is a Muckenhoupt A _p weight for 1 < p < ∞, then {pipe}{pipe}P ^* _Ff{pipe}{pipe} _{p, ω} ≤ C(F, w, p){pipe}{pipe}f{pipe}{pipe} _{p, ω}.