Higher order Journe commutators and characterizations of multi-parameter BMO

TL;DR

This paper characterizes boundedness of iterated commutators involving tensor products of Hilbert or Riesz transforms using mixed BMO classes, extending classical results to multi-parameter settings with operator-theoretic methods.

Contribution

It introduces new characterizations of multi-parameter BMO spaces via higher order Journe commutators, generalizing previous single-parameter results with novel operator constructions.

Findings

Established mixed BMO classes characterizing boundedness in $L^p$

Constructed Journé operators modeling multi-parameter Hilbert transforms

Proved upper estimates and weak factorization for commutators

Abstract

We consider iterated commutators of multiplication by a symbol function and tensor products of Hilbert or Riesz transforms. We establish mixed BMO classes of symbols that characterize boundedness of these objects in $L^p$. Little BMO and product BMO, big Hankel operators and iterated commutators are the base cases of our results. We use operator theoretical methods and existing profound results on iterated commutators for the Hilbert transform case, while the general result in several variables is obtained through the construction of a Journ\'e operator that models the behavior of the multiple Hilbert transform. Upper estimates for commutators with paraproduct free Journ\'e operators as well as weak factorisation results are proven.