Weighted estimates for dyadic paraproducts and t-Haar multipiers with complexity $(m,n)$

We extend the definitions of dyadic paraproduct and $t$-Haar multipliers to dyadic operators that depend on the complexity $(m,n)$, for $m$ and $n$ positive integers. We will use the ideas developed by Nazarov and Volberg to prove that the weighted $L^2(w)$-norm of a paraproduct with complexity $(m,n)$ associated to a function $b\in BMO$, depends linearly on the $A_2$-characteristic of the weight $w$, linearly on the $BMO$-norm of $b$, and polynomially in the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct (the one with complexity $(0,0)$). Also we prove that the $L^2$-norm of a $t$-Haar multiplier for any $t$ and weight $w$ depends on the square root of the $C_{2t}$-characteristic of $w$ times the square root of the $A_2$-characteristic of $w^{2t}$ and polynomially in the complexity $(m,n)$, recovering a result of Beznosova for the $(0,0)$-complexity case.