  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.
