The linear bound for Haar multiplier paraproducts

We study the natural resolution of the conjugated Haar multiplier $T_{\sigma }$ : $M_{w^{\frac{1}{2}}}{T}_{\sigma }{M}_{w^{-\frac{1}{2}}}=({\mathsf{P}}_{\widehat{w^{\frac{1}{2}}}}^{(0,1)}+{\mathsf{P}}_{\widehat{w^{\frac{1}{2}}}}^{(1,0)}+{\mathsf{P}}_{⟨w^{\frac{1}{2}}⟩}^{(0,0)})T_{\sigma }({\mathsf{P}}_{\widehat{w^{-\frac{1}{2}}}}^{(0,1)}+{\mathsf{P}}_{\widehat{w^{-\frac{1}{2}}}}^{(1,0)}+{\mathsf{P}}_{⟨w^{-\frac{1}{2}}⟩}^{(0,0)}),$ where each $M_{w^{\pm \frac{1}{2}}}$ is decomposed into its canonical paraproduct decomposition. We prove that each constituent operator obtained from this resolution has a linear bound on $L^2({\mathbb{R}}^d;w)$ 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 $L^2(w).$