A characterization of product BMO by commutators
Michael Lacey, Sarah Ferguson
Abstract
Let b be a function on the plane. Let Hj, j=1,2, be the Hilbert transform acting on the j-th coordinate on the plane. We show that the operator norm of the double commutator [[ Mb, H1], H2] is equivalent to the Chang-Fefferman BMO norm of b. Here, Mb denotes the operator which is multiplication by b. This result extends a well known theorem of Nehari on weak factorization in the Hardy space H1 to the same theorem on H1 of a product domain. The product setting is more delicate because of the presence of a two parameter family of dilations. The method of proof depends upon (a) A dyadic decomposition of product BMO by wavelets (b) a prior estimate of Ferguson and Sadosky involving rectangular BMO (c) and a careful control of certain measures related to those of Carleson.
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