Generalized factorization in Hardy spaces

Every inner function ϕ is associated with a factorization of Hardy space functions. This generalizes the classical Riesz factorization when ϕ( z) = z. In this generalized factorization, a function f ∈ Hp( D ) can be represented as a product of a ϕ--p inner and a ϕ--p outer function. We show that for certain functions f, the ϕ--p outer part has the same smoothness as the original function f. We show that if we pose certain conditions the ϕ--p inner factor is bounded. We also construct an Hinfinity function whose ϕ--p inner factor is not in any Hq for q > p.