Representation of bi-parameter singular integrals by dyadic operators

TL;DR

This paper establishes a dyadic representation theorem for bi-parameter singular integrals, expressing them as averages of bi-parameter shifts, and introduces a new product space T1 theorem.

Contribution

It provides a novel dyadic representation framework for bi-parameter singular integrals and a new version of the product space T1 theorem.

Findings

Representation of bi-parameter singular integrals as averages of shifts

Development of a new product space T1 theorem

Enhanced understanding of bi-parameter operator structure

Abstract

We prove a dyadic representation theorem for bi-parameter singular integrals. That is, we represent certain bi-parameter operators as rapidly decaying averages of what we call bi-parameter shifts. A new version of the product space T1 theorem is established as a consequence.