Simon Reinwand (Uniwersitet w Würzburgu)
On some Disparities of Multiplication and Composition Operators in BV Spaces
Although multiplication and composition of two functions are first grade operations, the corresponding multiplication and (autonomous) composition operator exhibit many weird and difficult to handle properties in BV spaces. We give criteria for acting conditions, as well as for injectivity, surjectivity, bijectivity and compactness in BV spaces. This leads naturally to a comprehensive study of multiplier sets, of which we will give a short overview. While most of them are easy to determine, some of them - especially those related to classical functions like Darboux and continuous functions - are unknown to our knowledge.
Moreover, we introduce a new type of convergence for composition operators in BV which leads to new a new proof for old continuity criteria. We hope that those ideas can be generalized to other BV-type spaces and give the relevant ideas and conjectures.
Apart from recalling known and discussing new results we put a particular emphasis on examples and counter examples.