Bilinear Calder\'on-Zygmund theory on product spaces

TL;DR

This paper develops a comprehensive bilinear Calderón-Zygmund theory on product spaces, establishing estimates, commutator decompositions, and bounds for bilinear singular integrals with applications to weighted and mixed-norm spaces.

Contribution

It introduces a general dyadic representation theorem for bilinear bi-parameter singular integrals and extends commutator estimates to the full range, simplifying previous proofs.

Findings

Established $L^p imes L^q o L^r$ estimates in full range

Developed commutator and iterated commutator bounds

Proved lower bounds using median method

Abstract

We develop a wide general theory of bilinear bi-parameter singular integrals $T$. First, we prove a dyadic representation theorem starting from $T1$ assumptions and apply it to show many estimates, including $L^p \times L^q \to L^r$ estimates in the full natural range together with weighted estimates and mixed-norm estimates. Second, we develop commutator decompositions and show estimates in the full range for commutators and iterated commutators, like $[b_1,T]_1$ and $[b_2, [b_1, T]_1]_2$, where $b_1$ and $b_2$ are little BMO functions. Our proof method can be used to simplify and improve linear commutator proofs, even in the two-weight Bloom setting. We also prove commutator lower bounds by using and developing the recent median method.