The Space of Bounded p(⋅)-Variation in the Sense Wiener-Korenblum with Variable Exponent

Abstract

In this paper we present the notion of the space of bounded p(⋅)-variation in the sense of Wiener- Korenblum with variable exponent. We prove some properties of this space and we show that the composition operator H, associated with h : R → R , maps the kBVW ([a,b]) pBV a b ⋅ κ , into itself, if and only if h is locally Lipschitz. Also, we prove that if the composition operator generated by h : [a,b]× →  maps this space into itself and is uniformly bounded, then the regularization of h is affine in the second variable, i.e. satisfies the Matkowski’s weak condition.