The Composition Operator and the Space of the Functions of Bounded Variation in Schramm-Korenblum's Sense

Abstract

We show that the composition operator H, associated with h:[a,b]→ℝ, maps the spaces Lip[a,b] on to the space κBVϕa,b of functions of bounded variation in Schramm-Korenblum's sense if and only if h is locally Lipschitz. Also, verify that if the composition operator generated by h:[a,b]×ℝ→ℝ maps this space into itself and is uniformly bounded, then regularization of h is affine in the second variable.