JournalszaaVol. 35, No. 3pp. 285–308

# On Superposition Operators in Spaces of Regular and of Bounded Variation Functions

• ### Artur Michalak ## Abstract

For a function $f:[0,1]\times \mathbb R \to \mathbb R$ we define the superposition operator $\Psi_f: \mathbb R^{[0,1]} \to \mathbb R^{[0,1]}$ by the formula $\Psi_f(\varphi)(t)=f(t,\varphi(t))$. 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

(1) it maps bounded subsets of $BV(0,1)$ into bounded sets if additionally $f$ is locally bounded,

(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

(3) $\limsup_{n\to \infty} n^{\frac{1}{2}}|f(t_n,x_n)-f(s_n,x_n)|<\infty$ 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.

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.