Weighted Inequalities for $t$-Haar multipliers

Abstract

In this paper, we provide necessary and sufficient conditions on a triple of weights $(u,v,w)$ so that the $t$-Haar multipliers $T_{w,\ddot{I}ƒ}^t$, $t\in \mathbb{R}$, %defined in \cite{P} when $\ddot{I}ƒ=1$, are uniformly (on the choice of signs $\ddot{I}ƒ$) bounded from $L^2(u)$ into $L^2(v)$. These dyadic operators have symbols $s(x,I)=\ddot{I}{ƒ}_I(w(x)/⟨w{⟩}_I{)}^t$ which are functions of the space variable $x\in \mathbb{R}$ and the frequency variable $I\in \mathcal{D}$, making them dyadic analogues of pseudo-differential operators. Here $\mathcal{D}$ denotes the dyadic intervals, $\ddot{I}{ƒ}_I=\pm 1$, and $⟨w{⟩}_I$ 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 $\ddot{I}{ƒ}_I\equiv 1$ 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.