On two weight estimates for dyadic operators

Abstract

We provide a quantitative two weight estimate for the dyadic paraproduct ${\pi }_b$ under certain conditions on a pair of weights $(u;v)$ and $b$ in $Car{l}_{u,v}$, a new class of functions that we show coincides with BMO when $u=v\in A_2^d$. We obtain quantitative two weight estimates for the dyadic square function and the martingale transforms under the assumption that the maximal function is bounded from $L_2(u)$ into $L_2(v)$ and $v\in R{H}_1^d$. Finally we obtain a quantitative two weight estimate from $L_2(u)$ into $L_2(v)$ for the dyadic square function under the assumption that the pair $(u;v)$ is in joint $A_2^d$ and $u^{-1}\in R{H}_1^d$, this is sharp in the sense that when $u=v$ the conditions reduce to $u\in A_2^d$ and the estimate is the known linear mixed estimate.