For a given single- or multivalued function and "atoms'' , let be the set of all measurable selections of the function which are constant on each . Continuity and differentiability of such operators are studied in spaces of measurable functions containing ideal, Orlicz and 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 in %%% here alterationspaces in the second part of this article [Z. Anal. Anwend. 31 (2011) (to appear)].
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Martin Väth, Continuity and Differentiability of Multivalued Superposition Operators with Atoms and Parameters I. Z. Anal. Anwend. 31 (2012), no. 1, pp. 93–124DOI 10.4171/ZAA/1450