Weighted estimates for dyadic paraproducts and -Haar multipliers with complexity

We extend the definitions of dyadic paraproduct and t-Haar multipliers to dyadic operators that depend on the complexity (m; n), for m and n natural numbers. We use the ideas developed by Nazarov and Volberg to prove that the weighted L2(w)-norm of a paraproduct with complexity (m; n), associated to a function b ∈ BMOd, depends linearly on the Ad/2-characteristic of the weight w, linearly on the BMOd-norm of b, and polynomially on the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct due to Beznosova. We also prove that the L2-norm of a t-Haar multiplier for any t ∈ R and weight w is a multiple of the square root of the Cd/2t-characteristic of w times the square root of the Ad/2-characteristic of w2t, and is polynomial in the complexity.