Decomposition theorems for Hardy spaces on convex domains of finite type

In this paper we study the holomorphic Hardy space Hp(Ω), where Ω is a smoothly bounded convex domain of finite type in ℂn. We show that for 0 < p ≤ 1, Hp(Ω) admits an atomic decomposition. More precisely, we prove that each f ∈ Hp(Ω) can be written as f = PS(∑j=0∞ νjaj) = ∑j=0;∞νjPS(aj ) where PS is the Szegö projection, the aj's are real variable p-atoms on the boundary ∂Ω, and the coefficients νj satisfy the condition ∑j=0∞ |νj|p ≲ ∥f∥H(p)(Ω)p. Moreover, we prove the following factorization theorem. Each f ∈ Hp(Ω) can be written as f = ∑j=0∞fjgj, where fj ∈ H2p, gj ∈ H2p, and ∑j=0∞∥fj∥H(2p) ∥gj∥H(2p) ≲ ∥f∥H(p)(Ω). Finally, we extend these theorems to a class of domains of finite type that includes the strongly pseudoconvex domains and the convex domains of finite type.