A characterization of product BMO by commutators
Abstract
Let b be a function on the plane. Let H_j, 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 [[ M_b, H_1], H_2] is equivalent to the Chang-Fefferman BMO norm of b. Here, M_b denotes the operator which is multiplication by b. This result extends a well known theorem of Nehari on weak factorization in the Hardy space H^1 to the same theorem on H^1 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.
Cite
@article{arxiv.math/0104144,
title = {A characterization of product BMO by commutators},
author = {Michael Lacey and Sarah Ferguson},
journal= {arXiv preprint arXiv:math/0104144},
year = {2007}
}
Comments
21 pages. To appear in Acta Math. Appendix corrected