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

Abstract

We show that the composition operatorH, associated withh:[a,b]→ℝ, maps the spacesLip[a,b]on to the spaceκBVϕa,bof functions of bounded variation in Schramm-Korenblum's sense if and only ifhis locally Lipschitz. Also, verify that if the composition operator generated byh:[a,b]×ℝ→ℝmaps this space into itself and is uniformly bounded, then regularization ofhis affine in the second variable.