zbMATH — the first resource for mathematics

On the continuity of the generalized Nemytskiĭ operator in spaces of differentiable functions. (English. Russian original) Zbl 1026.47050
Math. Notes 71, No. 2, 154-165 (2002); translation from Mat. Zametki 71, No. 2, 168-181 (2002).
The author gives sufficient conditions for the continuity of general nonlinear superposition operators (generalized Nemytskii operator) acting from the space \(C^m(\overline\Omega)\) of the differentiable functions on a bounded domain \(\Omega\) to 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 the order \(m\) inclusively. 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.

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: DOI