Linear bound for the dyadic paraproduct on weighted Lebesgue space [formula omitted

Abstract

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 that the bound on the norm of the dyadic paraproduct in the weighted Lebesgue space L 2 ( w ) depends linearly on the A 2 d characteristic of the weight w using Bellman function techniques and extrapolate this result to the L p ( w ) case.