Weighted Inequalities for t-Haar Multipliers

In this paper, we provide necessary and sufficient conditions on a triple of weights (u, v, w) so that the t-Haar multipliers Tw,σt, t∈R, are uniformly (on the choice of signs σ) bounded from L2(u) into L2(v). These dyadic operators have symbols s(x,I)=σI(w(x)/⟨w⟩I)t which are functions of the space variable x∈R and the frequency variable I∈D, making them dyadic analogues of pseudo-differential operators. Here D denotes the dyadic intervals, σI=±1, and ⟨w⟩I denotes the integral average of w on I. When w≡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 σI≡1 and u=v≡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.