A T(1) theorem for entangled multilinear dyadic Calder\'{o}n-Zygmund operators

TL;DR

This paper establishes a T(1) theorem for a broad class of dyadic multilinear operators acting on two-dimensional functions, extending classical Calderón-Zygmund theory to more complex, entangled structures.

Contribution

It introduces a boundedness criterion for entangled multilinear dyadic operators, generalizing classical results to operators with functions sharing variables, motivated by bilinear Hilbert transforms and ergodic averages.

Findings

Proves a T(1) theorem for entangled multilinear dyadic operators

Extends Calderón-Zygmund theory to more complex multilinear forms

Provides tools for analyzing bilinear Hilbert transforms and ergodic averages

Abstract

We prove a boundedness criterion for a class of dyadic multilinear forms acting on two-dimensional functions. Their structure is more general than the one of classical multilinear Calder\'{o}n-Zygmund operators as several functions can now depend on the same one-dimensional variable. The study of this class is motivated by examples related to the two-dimensional bilinear Hilbert transform and to bilinear ergodic averages. This paper is a sequel to the prior paper arXiv:1108.0917 by the first author.