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The Hilbert transform has a linear bound in the $A_{2}$ characteristic onweighted $L^{2}$, \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 constituentoperators in the natural weighted resolution of the conjugation$M_{w^{\frac{1}{2}}}\mathcal{S }M_{w^{-\frac{1}{2}}}$ induced by the canonicaldecomposition of a multiplier into paraproducts:% \begin{equation*}M_{f}=P_{f}^{-}+P_{f}^{0}+P_{f}^{+}. \end{equation*} The main tools used arecomposition of paraproducts, a product formula for Haar coefficients, theCarleson Embedding Theorem, and the linear bound for the square function.

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