Weighted estimates for dyadic paraproducts and $\mathbf{t}$- multipliers with complexity $(m,n)$

Abstract

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 $L^2(w)$\guio{norm} of a paraproduct with complexity~$(m,n)$, associated to a function $b\in \mathit{BMO}^d$, depends linearly on the $A^d_2$-characteristic of the weight~$w$, linearly on the $\mathit{BMO}^d$-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 $L^2$-norm of a $t$-Haar multiplier for any $t\in\mathbb{R}$ and weight~$w$ is a multiple of the square root of the $C^d_{2t}$-characteristic of $w$ times the square root of the $A^d_2$-characteristic of $w^{2t}$, and is polynomial in the complexity.