HIGHER ORDER COMMUTATORS AND MULTI-PARAMETER BMO

Abstract

In this article we highlight the interplay ofmulti-parameter BMO spaces and boundedness of corresponding commutators. In a variety of settings, we discuss two-sided norm estimates for commutators of classical singular operators with a symbol function. In its classical form, this concerns a theorem by Nehari, factorisation of Hardy space, Hankel and Toeplitz forms. We highlight recent results in which a characterization of Lp boundedness of iterated commutators of multiplication by a symbol function and tensor products of Riesz and Hilbert transforms is obtained, completing a theory on characterisation of BMO spaces begun by Cotlar, Ferguson and Sadosky. In the light of real analysis, we discuss results in a more intricate situation; commutators of multiplication by a symbol function and Calderón-Zygmund or 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 Hilbert or Riesz transforms or their tensor products are a representative testing class for Calderón-Zygmund or Journé operators.