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 $T^t_{w,\sigma}$, $t\in \R$, %defined in \cite{P} when $\sigma=1$, are uniformly (on the choice of signs $\sigma$) bounded from $L^2(u)$ into $L^2(v)$. These dyadic operators have symbols $s(x,I)=\sigma_I\,(w(x)/\langle w\rangle_I)^t$ 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, $\sigma_I=\pm1$, 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 $\sigma_I\equiv 1$ 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.