Linear bound for the dyadic paraproduct on weighted Lebesgue space L2(w)

The dyadic paraproduct is bounded in weighted Lebesgue spaces Lp(w) if and only if the weight w belongs to the Muckenhoupt class . However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest L2(w) case. In this paper we prove that the bound on the norm of the dyadic paraproduct in the weighted Lebesgue space L2(w) depends linearly on the characteristic of the weight w using Bellman function techniques and extrapolate this result to the Lp(w) case.