Chapter 8 - The superposition operator in spaces of smooth functions

Summary

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 C^k 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 C^k whose k-th. derivatives belong to the Hölder space H_φ. 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.