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 → :  , maps the () [ ] () W p BV 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.

On the generalized Wiener bounded variation spaces with p-variable

“…Also, there is defined the analog of absolute p-continuous functions in the framework of variable space and proved that this space of absolutely continuous functions is closed and separable subspace of BV p(•) [a; b]. For the different results concerning the space BV p(•) [a; b], and other spaces of bounded variation with p-variable see [5], [9], [10].…”