On superposition operators in spaces of regular and of bounded variation functions (Q330248)

Summary: For a function \(f:[0,1]\times \mathbb R \rightarrow \mathbb R\) we define the superposition operator \(\Psi_f: \mathbb R^{[0,1]} \rightarrow \mathbb R^{[0,1]}\) by the formula \(\Psi_f(\phi)(t)=f(t,\phi(t))\). First we provide necessary and sufficient conditions for \(f\) under which the operator \(\Psi_f\) maps the space \(R(0,1)\), of all real regular functions on \([0,1]\), into itself. Next we show that if an operator \(\Psi_f\) maps the space \(BV(0,1)\), of all real functions of bounded variation on \([0,1]\), into itself, then {\parindent=6mm \begin{itemize}\item[(1)] it maps bounded subsets of \(BV(0,1)\) into bounded sets if additionally \(f\) is locally bounded, \item[(2)] \(f= f_{cr}+f_{dr}\) where the operator \(\Psi_{f_{cr}}\) maps the space \(D(0,1)\cap BV(0,1)\), of all right-continuous functions in \(BV(0,1)\), into itself and the operator \(\Psi_{f_{dr}}\) maps the space \(BV(0,1)\) into its subset consisting of functions with countable support \item[(3)] \(\limsup_{n\rightarrow \infty} n^{\frac{1}{2}}|f(t_n,x_n)-f(s_n,x_n)|<\infty\) for every bounded sequence \((x_n)\subset \mathbb R\) and for every sequence \(([s_n,t_n))\) of pairwise disjoint intervals in \([0,1]\) such that the sequence \((|f(t_n,x_n)-f(s_n,x_n)|)\) is decreasing. \end{itemize}} Moreover we show that if an operator \(\Psi_f\) maps the space \(D(0,1)\cap BV(0,1)\) into itself, then \(f\) is locally Lipschitz in the second variable uniformly with respect to the first variable.