On two weight estimates for dyadic operators

TL;DR

This paper establishes quantitative two-weight estimates for dyadic operators such as paraproducts, square functions, and martingale transforms under specific weight conditions, extending known results and introducing new classes of functions.

Contribution

It introduces a new class of functions $Carl_{u,v}$ and provides sharp two-weight estimates for dyadic operators under various weight conditions, generalizing previous results.

Findings

Quantitative estimates for dyadic paraproducts under new weight conditions.

Sharp two-weight bounds for dyadic square functions and martingale transforms.

Identification of a new function class $Carl_{u,v}$ coinciding with BMO when weights are equal.

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 $Carl_{u,v}$, a new class of functions that we show coincides with BMO when $u = v \in A^d_2$. 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 $v \in RH^d_1$. 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 $A^d_2$ and $u^{-1} \in RH^d_1$, this is sharp in the sense that when $u = v$ the conditions reduce to $u \in A^d_2$ and the estimate is the known linear mixed estimate.