A lower bound criterion for iterated commutators

We consider iterated commutators of multiplication by a symbol function and smooth Calderón-Zygmund operators, described by Fourier multipliers of homogeneity 0. We establish a criterion for a collection of symbols so that the corresponding Calderón-Zygmund operators characterize product BMO by means of iterated commutators. We therefore extend, in part, the line of one-parameter results following the work of Uchiyama and Li as well as the result in several parameters, concerning commutators with Riesz transforms by Lacey, Petermichl, Pipher, Wick.