The linear bound for Haar multiplier paraproducts

We study the natural resolution of the conjugated Haar multiplier T-sigma: M-w1/2 T sigma Mw-1/2 = (p(0,1)/w(1/2) + p(1,0)/w(1/2) + p(0,0)/< w(1/2)>) T-sigma (p(0,1)/w(-1/2) + p(1,0)/w(-1/2) + p(0,0)/< w(-1/2)>). where each M-w +/- 1/2 is decomposed into its canonical paraproduct decomposition. We prove that each constituent operator obtained from this resolution has a linear bound on L-2 (R-d; w) in terms of the A(2) characteristic of w. The main tools used are a "product formula" for Haar coefficients, the Carleson Embedding Theorem, the linear bound for the square function, and the well-known linear bound of T-sigma on L-2 (w).