On some properties of the superposition operator in generalized Orlicz spaces of vector-valued functions. (English) Zbl 0608.47068

The author proves some continuity and boundedness results for the superposition operator \(Fx(s)=f(s,x(s))\) in Orlicz spaces which are parallel to those given by M. A. Krasnosel’skij and Ya. B. Rutitskij in their book ”Convex functions and Orlicz spaces” (1961; Zbl 0095.091), with two important generalizations: first, the functions x are allowed to take values in a separable reflexive Banach space; second, the Young function which generates the Orlicz space is ”variable” (i.e. non- autonomous), and thus one has to use the general setting of modular spaces [see e.g. J. Musielak, Orlicz spaces and modular spaces, Lect. Notes Math. 1034 (1983; Zbl 0557.46020)].
Reviewer’s remark: Further boundedness results for the superposition operator are given by the same author in Bull. Pol. Acad. Sci. Math. 33, 531-540 (1985; Zbl 0587.46027).
Reviewer: J.Appell


47H99 Nonlinear operators and their properties
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)