Bilinear representation theorem

TL;DR

This paper introduces a new way to represent bilinear Calderón-Zygmund operators as sums of simpler dyadic operators, leading to a sparse T1 theorem for bilinear singular integrals.

Contribution

It provides a novel representation theorem for bilinear operators and establishes a sparse bound, advancing the understanding of bilinear Calderón-Zygmund theory.

Findings

Representation of bilinear operators as sums of dyadic operators

Proof of sparse bounds for these dyadic operators

Derivation of a sparse T1 theorem for bilinear singular integrals

Abstract

We represent a general bilinear Calder\'on-Zygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so called sparse T1 theorem for bilinear singular integrals.