Characterization of compactness of commutators of bilinear singular integral operators

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.