Higher order Journé commutators and characterizations of multi-parameter BMO

We characterize LpLp boundedness of iterated commutators of multiplication by a symbol function and tensor products of Riesz and Hilbert transforms. We obtain a two-sided norm estimate that shows that such operators are bounded on LpLp if and only if the symbol belongs to the appropriate multi-parameter BMO class. We extend our results to a much more intricate situation; commutators of multiplication by a symbol function and paraproduct-free Journé operators. We show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi-parameter BMO class. In this sense the tensor products of Riesz transforms are a representative testing class for Journé operators.Previous results in this direction do not apply to tensor products and only to Journé operators which can be reduced to Calderón–Zygmund operators. Upper norm estimates of Journé commutators are new even in the case of no iterations. Lower norm estimates for iterated commutators only existed when no tensor products were present. In the case of one dimension, lower estimates were known for products of two Hilbert transforms, and without iterations. New methods using Journé operators are developed to obtain these lower norm estimates in the multi-parameter real variable setting.