Composition operators acting on Sobolev spaces of fractional order – a survey on sufficient and necessary conditions

What follows is a survey of recent results on sufficient and necessary conditions on composition operators to map one Sobolev space of fractional order into another. This report may be taken as a continuation of the contribution of G.Bourdaud given at the forerunner conference of this one, held in Friedrichroda 1992, cf. [Bo 5]. Composition operators are simple examples of nonlinear operators. Surprisingly enough, the behaviour of those operators even on Sobolev spaces itself is not completely understood up to now. As we hope to show, it turns out that