On Connections Between Delta-Convex Mappings and Convex Operators

We study conditions under which every delta-convex (d.c.) mapping is the difference of two continuous convex operators, and vice versa. In particular, we prove that each d.c. mapping F:(a,b)→ Y is the difference of two continuous convex operators whenever Y belongs to a large class of Banach lattices which includes all Lᵖ(μ) spaces (1≤ p≤∞). The proof is based on a result about Jordan decomposition of vector-valued functions. New observations on Jordan decomposition of finitely additive vector-valued measures are also presented.