Continuity and Differentiability of Multivalued Superposition Operators with Atoms and Parameters I

Abstract

For a given single- or multivalued function $f$ and “atoms” $S_i$, let $S_f(\lambda ,x)$ be the set of all measurable selections of the function $s\mapsto f(\lambda ,s,x(s))$ which are constant on each $S_i$. Continuity and differentiability of such operators are studied in spaces of measurable functions containing ideal, Orlicz and $L_p$ spaces with new results for the parameter-dependent case even for single-valued superposition operators without atoms. A motivation is to apply the results for variant of such maps $S_f$ in Sobolev spaces in the second part of this article [Z. Anal. Anwend. 31 (2011) (to appear)].