Off-diagonal estimates for bilinear commutators

TL;DR

This paper characterizes the boundedness of bilinear commutators on product spaces using testing conditions on the multiplying function, extending to multilinear operators and covering nearly all integrability exponents.

Contribution

It introduces a minimal non-degeneracy condition and identifies testing criteria for the boundedness of bilinear commutators, extending results to multilinear operators.

Findings

Characterization of boundedness via testing conditions

Extension to multilinear operators

Almost complete coverage of integrability exponents

Abstract

We find a minimal notion of non-degeneracy for bilinear singular integral operators $T$ and identify testing conditions on the multiplying function $b$ that characterize the $L^p\times L^q\to L^r,$ $1<p,q<\infty$ and $r>\frac{1}{2},$ boundedness of the bilinear commutator $[b,T]_1(f,g) = bT(f,g) - T(bf,g).$ Our arguments cover almost all arrangements of the integrability exponents $p,q,r,$ with a single open problem presented in the end. Additionally, the arguments extend to the multilinear setting.