On the Continuity of the Generalized Nemytskii Operator on Spaces of Differentiable Functions

We obtain sufficient conditions for the continuity of the general nonlinear superposition operator (generalized Nemytskii operator) acting from the space $C^m(\overline \Omega)$ of differentiable functions on a bounded domain $\Omega$ to the Lebesgue space $L_p(\Omega)$. The values of operators on a function $u\in C^m(\overline \Omega)$ are locally determined by the values of both the function $u$ itself and all of its partial derivatives up to order $m$ inclusive. In certain particular cases, the sufficient conditions obtained are proved to be necessary as well. The results are illustrated by several examples, and an application to the theory of Sobolev spaces is also given.