Weighted Inequalities for $t$-Haar multipliers

TL;DR

This paper characterizes two-weight and one-weight boundedness criteria for the $t$-Haar multipliers $T^t_{w,\sigma}$ acting on $L^2(u)$ to $L^2(v)$, uniformly in the sign sequence $\sigma$. It introduces four sharp conditions $(i)$–$(iv)$ involving the weights $(u,v,w)$ and the parameter $t$, establishing that their satisfaction is equivalent to uniform boundedness, with a quantitative bound $\|T^t_{w,\sigma}\|_{L^2(u)\to L^2(v)} \approx \sqrt{C_1}+\sqrt{C_2}+\sqrt{C_3}+C_4$. The work extends Nazarov–Treil–Volberg martingale-transform theory to dyadic variable-coefficient Haar multipliers, develops Sawyer-type testing for individual operators, and provides reductions to simpler one-weight and unweighted cases. The results apply in spaces of homogeneous type and yield a comprehensive framework for two-weight and testing theories of $t$-Haar multipliers, with implications for dyadic pseudo-differential analogues. Overall, the paper advances understanding of how three-weight and Carleson-type conditions govern boundedness in this dyadic multiplier setting.

Abstract

In this paper, we provide necessary and sufficient conditions on a triple of weights $(u,v,w)$ so that the $t$-Haar multipliers $T^t_{w,σ}$, $t\in \R$, %defined in \cite{P} when $σ=1$, are uniformly (on the choice of signs $σ$) bounded from $L^2(u)$ into $L^2(v)$. These dyadic operators have symbols $s(x,I)=σ_I\,(w(x)/\langle w\rangle_I)^t$ which are functions of the space variable $x\in\R$ and the frequency variable $I\in \mathcal{D}$, making them dyadic analogues of pseudo-differential operators. Here $\mathcal{D}$ denotes the dyadic intervals, $σ_I=\pm1$, and $\langle w\rangle_I$ denotes the integral average of $w$ on $I$. When $w\equiv 1$ we have the martingale transform and our conditions recover the known two-weight necessary and sufficient conditions of Nazarov, Treil and Volberg. %We will discuss some relations between the three weights inequality for these operators given the inequality for other dyadic operators. We also show how these conditions are simplified when $u=v$. In particular, the martingale one-weight and the $t$-Haar multiplier unsigned and unweighted (corresponding to $σ_I\equiv 1$ and $u=v\equiv 1$) known results are recovered or improved. We also obtain necessary and sufficient testing conditions of Sawyer type for the two-weight boundedness of a single variable Haar multiplier similar to those known for the martingale transform.

Theorems & Definitions (30)

• Theorem 1: Two-weight theorem
• Theorem 2: One-weight theorem
• Lemma 1: NTV1
• Lemma 2
• Lemma 3: Weighted Carleson's Lemma
• Remark 1
• Proposition 1
• Lemma 4: Buckley's characterization of $RH^d_1$ BuBeRez
• Lemma 5: Katz-Pereyra '99, Lemma 1
• proof : Proof of Proposition \ref{['prop:RHp']}