In this paper we present a necessary condition for an autonomous superposition operator to act in the space of functions of Waterman-Shiba bounded variation. We also show that if a (general) superposition operator applies such space into itself and it is uniformly bounded, then its generating function satisfies a weak Matkowski condition.
In this paper we introduce the concept of bounded variation for functions defined on compact subsets of the complex plane \(\mathbb{C}\), based on the notion of variation along a curve as defined by Ashton and Doust; We describe in detail the space so generated and show that it can be equipped, in a natural way, with the structure of a Banach algebra. We also present a necessary condition for a composition operator \(C_\varphi\) to act between two such spaces.
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