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Théorèmes de bornage pour l’opérateur de Nemyckii dans les espaces idéaux. (Bounding theorems for Nemytskij operators in ideal spaces). (French) Zbl 0619.47051

Some classes of function spaces are described with the property that, whenever the nonlinear superposition operator \(Fx(s)=f(s,x(s))\) acts between such spaces, F is bounded on each ball. For example, an Orlicz space \(L_ M\) belongs to this class if and only if \(M\in \Delta_ 2\). As a further example, interpolation spaces of Lorentz and Marcinkiewicz type are considered. Finally, some relations with ”weak” Orlicz spaces are established.
Similar classes of spaces have been studied by W. M. Kozlowski [Ann. Ser. Math. Pol., Ser. I, Commentat. Math. 22, 85-103 (1980; Zbl 0471.47035)]. An essential generalization is given in a forthcoming paper of the first author and P. P. Zabrejko [Boundedness properties of the superposition operator, Bull. Polish Acad. Sci. (to appear)].

MSC:

47H99 Nonlinear operators and their properties
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46M35 Abstract interpolation of topological vector spaces

Citations:

Zbl 0471.47035
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