Two-weight estimates for the square function and $t$-Haar multipliers
Daewon Chung, Jean Carlo Moraes, María Cristina Pereyra, Brett Wick
TL;DR
The paper characterizes two-weight boundedness of the dyadic weighted square function S^d_w between L^2 spaces with different weights, giving explicit sufficient conditions on the triple (u,v,w) and, when w is dyadic doubling, necessary conditions. It introduces three-weight and restricted A_2-type conditions and uses weighted Haar expansions, Beznosova's Little Lemma, and Sawyer estimates to derive sharp upper bounds for ||S^d_w||_{L^2(u)→L^2(v)} that depend on the A_2-type quantity [uw^{-1},vw^{-1}]_{A_2^d(w)} and a Carleson constant controlling Δ^w_I(u^{-1}w). The work then connects these square-function bounds to the two-weight boundedness of signed t-Haar multipliers T^t_{w,σ}, showing that, under suitable hypotheses (e.g., w∈A_∞^d), the multiplier bounds are controlled by the square-function bounds and additional Carleson-type and resolvent-type conditions. By recovering classical results (NTV99, Buckley, Per94) as corollaries, the paper situates the two-weight theory within the broader Fefferman–Kenig–Pipher framework and provides a unified approach to square functions and Haar multipliers in the dyadic setting, with implications for paraproduct resolvents and related harmonic analysis problems.
Abstract
We present necessary and sufficient conditions on triples of weights $(u,v,w)$ for the boundedness of the dyadic weighted square function $S_w$ from $L^2(u)$ into $L^2(v)$. We use this characterization to obtain necessary and sufficient conditions for the boundedness of the $t$-Haar multipliers from $L^2(u)$ into $L^2(v)$ in terms of boundedness of the dyadic weighted square function.