Regularity of the composition operator in spaces of Hölder functions (Q1576807)

This is a nice survey on various regularity properties of the composition operator \((f,g)\mapsto g\circ f\) which unifies the (linear) inner substitution operator \(g\mapsto g\circ f\) (for fixed \(f\)) and the (nonlinear) outer superposition operator \(f\mapsto g\circ f\) (for fixed \(g\)). The main emphasis is put on sufficient differentiability conditions (which are often also necessary or close to being necessary) for this operator between Hölder and Schauder spaces. Specialists in the field will find the detailed list of references useful. Surprisingly, the authors seem to be unaware of the contributions of \textit{M. Lanza de Cristoforis} [e.g., Indag. Math., New Ser. 5, No. 4, 457-482 (1994; Zbl 0822.46051) or \textit{M. L. Rend} Accad. Naz. Soc. XL, V. Ser., Mem. Mat. 15, No. 1, 93-109 (1991; Zbl 0829.47059)].