The Space of Bounded p(·)-Variation in Wiener’s Sense with Variable Exponent

Abstract

In this paper, we proof some properties of the space of bounded p(⋅)-variation in Wiener's sense. We show that a functions is of bounded p(⋅)-variation in Wiener's sense with variable exponent if and only if it is the composition of a bounded nondecreasing functions and hölderian maps of the () p 1 ⋅ variable exponent. We show that the composition operator H, associated with h   : → , maps the spaces () [ ] () p WBV a b , ⋅ 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.

Functions of Bounded (<i>p</i>(⋅), 2)-Variation in De la Vallée Poussin-Wiener’s Sense with Variable Exponent

“…In [23] the authors present and study the space of functions of bounded ( ) ; , : sup , It is endowed with the functional:…”

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].…”

Locally Defined Operators and Locally Lipschitz Composition Operators in the Space <i>WBV</i><i>p</i>(·)([a, b])

“…The following structural theorem is taken from [2], this gives us a characterization of the members of…”

“…This paper lies in the field of variable exponent function spaces, exactly we will deal with the space able exponent (see [1], [2]). Variable exponent Lebesgue spaces appeared in the literature in 1931 in the paper by Orlicz [3].…”