Linear bound for the dyadic paraproduct on weighted Lebesgue space L2(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 that the bound on the norm of the dyadic paraproduct in the weighted Lebesgue space L-2(w) depends linearly on the Ad characteristic of the 2 weight w using Bellman function techniques and extrapolate this result to the L-p(w) case. (c) 2008 Elsevier Inc. All rights reserved.