Uniformly bounded composition operators on a Banach space of bounded Wiener-Young variation functions (Q2857010)

Consider a real interval \(I\), real Banach spaces \(X\), \(Y\), a convex function \(\varphi:[0,\infty) \to [0,\infty)\) such that \(\varphi(0)=0\), \(\varphi(t)>0\) for all \(t>0\), \(\varphi(t) \to \infty\) for \(t \to \infty\), and the Banach space \(BV_\varphi(I,X)\) of functions \(f:I\to X\) of bounded \(\varphi\)-variation in the sense of \textit{N. Wiener} [J. Math. Phys., MIT 3, 72--94 (1924; JFM 50.0203.01)] and \textit{L. C. Young} [C. R. Acad. Sci., Paris 204, No. 7, 470--472 (1937; Zbl 0016.10501; JFM 63.0182.03)]. Similarly for \(Y, \psi\). Let \(C\subset X\) be a convex set, \(\mathrm{int} C \neq \emptyset\). The main result of the present paper states that, if \(H\) is a uniformly bounded Nemytskij composition operator which maps the set of functions \(f\in BV_\varphi(I,X)\) with \(f(I)\subset C\) into \(BV_\psi(I,Y)\) and has a generator \(h\), i.e., \((Hf)(t)=h(t,f(t))\), then this generator is of the form \(h(t,x)=a(t)x+b(t)\). A~consequence concerning the functional equation \(f(t)=h(t,f(\alpha(t))\) with given \(h:I\times X \to X\) and \(\alpha:I \to I \) is mentioned.