Uniformly continuous composition operators in the space of bounded Φ-variation functions in the Schramm sense

Abstract

We prove that any uniformly continuous Nemytskii composition operator in the space of functions of bounded generalized \(\Phi\)-variation in the Schramm sense is affine. A composition operator is locally defined. We show that every locally defined operator mapping the space of continuous functions of bounded (in the sense of Jordan) variation into the space of continous monotonic functions is constant.

Keywords

• \(\Phi\)-variation in the sense of Schramm
• uniformly continuous operator
• regularization
• Jensen equation
• locally defined operators