Linear bound for the dyadic paraproduct on weighted Lebesgue space $L_2(w)$

The dyadic paraproduct is bounded in weighted Lebesgue spaces $L_p(w)$ if and only if the weight $w$ belongs to the Muckenhoupt class $A_p^d$. However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest $L_2(w)$ case. In this paper we prove the linear bound on the norm of the dyadic paraproduct in the weighted Lebesgue space $L_2(w)$ using Bellman function techniques and extrapolate this result to the $L_p(w)$ case.