Nonautonomous superposition operators in the spaces of functions of bounded variation

Abstract

The main goal of this paper is to give an answer to the main problem of the theory of nonautonomous superposition operators acting in the space of functions of bounded variation in the sense of Jordan. Namely, we give necessary and sufficient conditions which guarantee that nonautonomous superposition operators map that space into itself and are locally bounded. Moreover, special attention is drawn to nonautonomous superposition operators generated by locally bounded mappings as well as to superposition operators generated by functions with separable variables.

On some nonlinear operators in $${\varvec{\Lambda }} {\varvec{BV}}$$ Λ B V -spaces

“…The main result in this section is Theorem 7 which gives necessary and sufficient conditions for nonautonomous superposition operators to map a space of functions of bounded Λ-variation into itself. The results included in this section are mainly inspired by the paper [3]. However, let us emphasize the fact that nonautonomous superposition operators in the spaces of functions of bounded Λ-variation behave in a slightly different way than in spaces of functions of bounded variation in the sense of Jordan (see Example 1).…”

A survey on composition operators on some function spaces

“…The following result is a special case of Theorem 3.8 in [13] (when X = BV ϕ with the Young function ϕ(t) = t p ). (a) for every r > 0 there exists a constant M r > 0 such that for every k ∈ N, every finite partition…”

On integral bounded variation

“…The problem of acting conditions and related problems have been studied extensively for different concepts of variation (see the classical papers [17,20], more recent results [6,[9][10][11]21] and also [1] for an overview). Therefore, it seems to be important to answer this question also in this case.…”