Bellman functions and $L^p$ estimates for paraproducts

TL;DR

This paper provides an explicit Bellman function formula for $L^p$ bounds of dyadic paraproducts and applies it to give alternative proofs for classical operators like martingale and heat flow paraproducts.

Contribution

It introduces a new explicit Bellman function approach for $L^p$ estimates of paraproducts and extends its application to various classical operators.

Findings

Explicit Bellman function formula for dyadic paraproducts.

Alternative proofs for martingale and heat flow paraproduct estimates.

Unified approach to classical operator bounds using Bellman functions.

Abstract

We give an explicit formula for one possible Bellman function associated with the $L^p$ boundedness of dyadic paraproducts regarded as bilinear operators or trilinear forms. Then we apply the same Bellman function in various other settings, to give self-contained alternative proofs of the estimates for several classical operators. These include the martingale paraproducts of Ba\~{n}uelos and Bennett and the paraproducts with respect to the heat flows.