The Linear Bound for the Natural Weighted Resolution of the Haar Shift

The Hilbert transform has a linear bound in the $A_2$ characteristic on weighted $L^2$, \begin{equation*} \left\Vert H\right\Vert {L^{2}(w)\rightarrow L^{2}(w)}\lesssim \left[ w \right] {A{2}}, \end{equation*} and we extend this linear bound to the nine constituent operators in the natural weighted resolution of the conjugation $M_{w^{\frac{1}{2}}}\mathcal{S}{M}_{w^{-\frac{1}{2}}}$ induced by the canonical decomposition of a multiplier into paraproducts:% \begin{equation*} M{f}=P_{f}^{-}+P_{f}^{0}+P_{f}^{+}. \end{equation*} The main tools used are composition of paraproducts, a product formula for Haar coefficients, the Carleson Embedding Theorem, and the linear bound for the square function.