J ul 2 00 4 HANKEL OPERATORS IN SEVERAL COMPLEX VARIABLES AND PRODUCT BMO ( ⊗ n 1 C + )

H(⊗1 C+) denotes the Hardy space of square integrable functions analytic in each variable separately. Let P be the natural projection of L(⊗1 C+) onto H(⊗1 C+). A Hankel operator with symbol b is the linear operator from H(⊗1 C+) to H(⊗1 C+) given by Hbφ = Pbφ. We show that ‖Hb‖ ≃ ‖Pb‖BMO(⊗n 1 C+), where the right hand norm is S.-Y. Chang and R. Fefferman product BMO. This fact has well known equivalences in terms of commutators and the weak factorization ofH(⊗1 C+). In the case of two complex variables, this is due to Ferguson and Lacey [8]. While the current proof is inductive, and one can take the one complex variable case as the basis step, it is heavily influenced by the methods of Ferguson and Lacey. The induction is carried out with a particular form of a lemma due to Journé [10], which occurs implicitly in the work of J. Pipher [13].