Uniformly bounded composition operators in the Banach space of absolutely continuous functions (Q435069)

An operator \(H\) mapping a metric space \(\mathcal{Y}\) into a metric space \(\mathcal{Z}\) is said to beNEWLINENEWLINE(i) uniformly bounded if, for any \(t>0\), there is a nonnegative real number \(\gamma (t)\) such that, for any nonempty set \(B\subset \mathcal{Y}\), if \(\text{diam}B\leq t\), then \(\text{diam}H(B)\leq \gamma (t)\);NEWLINENEWLINE(ii) equidistantly uniformly bounded, if this condition holds true for all two-points sets \(B\).NEWLINENEWLINEThe authors prove that the generator \(h\) of any uniformly bounded or equidistantly uniformly bounded Nemytskij operator \(H\) mapping the set \(AC(I,J)\) (of absolutely continuous functions \(\varphi \in J^{I}\), where \(I, J\subset {\mathbb R}\) are intervals) into a Banach space \(AC(I,{\mathbb R})\) must be of the form NEWLINE\[NEWLINEh(x,y)=\alpha (x)y+\beta (x),\quad x\in I,\; y\in {\mathbb R},NEWLINE\]NEWLINE for some functions \(\alpha ,\beta \in AC(I,{\mathbb R})\). This improves the earlier result of the second author [``Uniformly continuous superposition operators in the space of differentiable functions and absolutely continuous functions'', Int. Ser. Numer. Math. 157, 395--404 (2008; \url{doi:10.1007/978-3-7643-8773-0_15})], where \(H\) is assumed to be uniformly continuous.NEWLINENEWLINEMoreover, it is proved that, if \(H : AC(I,J)\to AC(I,{\mathbb R})\) is a Nemytskij operator, then its corresponding generator is a continuous function.