The Linear Bound for Haar Multiplier Paraproducts

We study the natural resolution of the conjugated Haar multiplier $M_{w^{\frac{1}{2}}}{T}_{\sigma }{M}_{w^{-\frac{1}{2}}},$ where the multiplication operators $M_{w^{\pm \frac{1}{2}}}$ are decomposed into their canonical paraproduct decompositions. 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 $L^2({\mathbb{R}}^d,w)\mathrm{.}$