Factorization in Banach function spaces

AbstractLet ϱ be a function norm based on a σ-finite measure space (Ω,Σ,μ). The Hölder inequality implies that, given f in Lϱ and g in Lϱ, the product fg belongs to L1(μ) and satisfies |fg|1≤ϱ(f)ϱ′(g). It is proved that, if ϱ is saturated and has the Fatou property, then every function ϕ in L1(μ) can be factorized as ϕ = fg, with f in Lϱ, g in Lϱ′ and |ϕ|1=ϱ(f)ϱ′(g). If ϱ is saturated and has the Riesz-Fischer property, then every L1-function can still be factorized in this way, but the condition on the norms of the factors must be weakened. A further weakening of the conclusion is necessary for the case when ϱ is assumed merely to be saturated. In that case, such a factorization of an arbitrary L1-function may not be possible, although it is almost possible in a certain measure-theoretic sense