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

TL;DR

This paper establishes a linear bound for the dyadic paraproduct operator in weighted Lebesgue spaces $L_2(w)$, advancing understanding of its norm dependence on weights and extending results to $L_p(w)$ spaces.

Contribution

The paper proves the first linear bound on the dyadic paraproduct norm in $L_2(w)$ using Bellman function techniques and extends this to $L_p(w)$ spaces.

Findings

Proved linear bound for dyadic paraproduct in $L_2(w)$.

Extended the bound to $L_p(w)$ spaces.

Used Bellman function techniques for the proof.

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 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.