In this paper, we provide necessary and sufficient conditions on a triple of weights (u, v, w) so that the t-Haar multipliers \(T^t_{w,\sigma }\) , \(t\in \mathbb {R}\) , 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 \(x\in \mathbb {R}\) and the frequency variable \(I\in \mathcal {D}\) , making them dyadic analogues of pseudo-differential operators. Here \(\mathcal {D}\) denotes the dyadic intervals, \(\sigma _I=\pm 1\) , 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 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.
