In this chapter we study the superposition operator in various spaces of functions which are characterized by certain smoothness properties. We begin with a necessary and sufficient acting and continuity condition for F in the space Ck of k-times continuously differentiate functions. Surprisingly, without the continuity requirement for F the generating function f need not even be continuous. Afterwards, we show that a (global) Lipschitz condition for F is “never” satisfied, while a (local) Darbo condition holds “always”. This is in sharp contrast to the situation in spaces of measurable functions dealt with in Chapters 2 – 5, and also in the space C.

In the second part we try to develop a parallel theory in the spaces of all functions from Ck whose k-th. derivatives belong to the Hölder space Hφ. In particular, we give a sufficient acting and boundedness condition.

The last part is concerned with the superposition operator in various classes of smooth (i.e. C∞) functions, including Roumieu spaces, Beurling spaces, Gevrey spaces, and their projective and inductive limits. It turns out that an acting condition for the operator F in such classes, together with suitable additional growth conditions on the derivatives of the function f, guarantees not only the boundedness and continuity, but also the compactness of F.