Bellman function technique for multilinear estimates and an application to generalized paraproducts

TL;DR

This paper introduces a versatile Bellman function technique to establish L^p estimates for two-dimensional multilinear forms, generalizing classical paraproducts and twisted variants, with applications to generalized paraproducts.

Contribution

It develops a general Bellman function approach for dyadic multilinear operators and applies it to generalized paraproducts using combinatorics of integer partitions.

Findings

Established L^p estimates for a broad class of multilinear forms

Unified treatment of classical and twisted paraproducts

Demonstrated the method's effectiveness through applications to generalized paraproducts

Abstract

We prove L^p estimates for a class of two-dimensional multilinear forms that naturally generalize (dyadic variants of) both classical paraproducts and the twisted paraproduct introduced in [5] and studied in [1] and [6]. The method we use builds on the approach from [6] and we present it as a rather general technique for proving estimates on dyadic multilinear operators. In the particular application to "generalized paraproducts" this method is combined with combinatorics of integer partitions.