Characterization of compactness of commutators of bilinear singular integral operators

TL;DR

This paper investigates the conditions under which commutators of bilinear Calderón-Zygmund operators are compact, establishing that membership in $cmo$ is both necessary and sufficient for compactness.

Contribution

It proves that for certain non-degenerate bilinear Calderón-Zygmund operators, the symbol's membership in $cmo$ is necessary and sufficient for the commutator's compactness.

Findings

Commutators are bilinear compact operators when the symbol is in $cmo$.

Membership in $cmo$ is necessary and sufficient for compactness in certain cases.

The work characterizes the compactness of commutators in terms of symbol properties.

Abstract

The commutators of bilinear Calder\'on-Zygmund operators and point-wise multiplication with a symbol in $cmo$ are bilinear compact operators on product of Lebesgue spaces. This work shows that, for certain non-degenerate Calder\'on-Zygmund operators, the symbol being in $cmo$ is not only sufficient but actually necessary for the compactness of the commutators.