This article is an interesting review of results concerning autonomous and nonautonomous superposition operators in some function spaces such as, e.g., the space \(C([a,b])\) of all continuous functions \([a,b]\to \mathbb R\) with the maximum norm, the space \(C^{1}([a,b])\) of all continously differentiable functions \([a,b]\to \mathbb R\) with the norm \(\|f\|_{C^1}:=|f(a)|+\|f'\|_C\), and the space \(BV([a,b])\) of all functions \([a,b]\to \mathbb R\) of bounded variation in the sense of Jordan with the norm \(\|f\|_{BV}:=|f(a)|+\mathrm{Var}(f;[a,b])\).
The authors focus on mapping, continuity and boundedness properties of superposition operators in those spaces. Several interesting examples and counterexamples are provided.
Reviewer’s remarks. One can find the statement of Theorem 4.3 in [A. G. Ljamin, “On the acting problem for the Nemytskij operator in the space of functions of bounded variation” (Russian), 11th School Theory Oper. Function Spaces, Chel’jabinsk, 63–64 (1986)] (see the references in the paper under review), but without a proof. However, as has been shown in the paper “A counterexample to Ljamin’s theorem” by P. Maćkowiak [Proc. Am. Math. Soc., to appear], Theorem 4.3 is false. Let us also point out that inequality (4.2) is not true. It is enough to consider \(h:[0,1]\times \mathbb R\to\mathbb R\) , \(h(t,u)=\sin (t+u)\) and \(f:[0,1]\to [0,1]\), \(f(t)=t\).