Upper Bound for Multi-parameter Iterated Commutators

Abstract

We show that the product BMO space can be characterized by iterated commutators of a large class of Calder\'on-Zygmund operators. This result follows from a new proof of boundedness of iterated commutators in terms of the BMO norm of their symbol functions, using Hyt\"onen's representation theorem of Calder\'on-Zygmund operators as averages of dyadic shifts. The proof introduces some new paraproducts which have BMO estimates.