Factorization of Hardy spaces and Hankel operators on convex domains in ℂn

In this article we study the (small) Hankel operator hb on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂn. We completely characterize the Hankel operators hb that are bounded, compact, and belong to the Schatten ideal Sp, for 0 < p < ∞. In particular, if hb denotes the Hankel operator on the Hardy space H2(Ω), we prove that hb is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈ Bp, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H1 (Ω), a result that is of independent interest.