# On automatic boundedness of Nemytskiĭ set-valued operators

Studia Mathematica (1995)

• Volume: 113, Issue: 1, page 65-72
• ISSN: 0039-3223

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## Abstract

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Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let ${N}_{F}$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F:\Omega ×X\to {2}^{Y}$. It is shown that if ${N}_{F}$ maps a modular space $\left(N\left(L\left(\Omega ,\Sigma ,\mu ;X\right)\right),{\varrho }_{N,\mu }\right)$ into subsets of a modular space $\left(M\left(L\left(\Omega ,\Sigma ,\mu ;Y\right)\right),{\varrho }_{M,\mu }\right)$, then ${N}_{F}$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that ${r}_{K}=sup{\varrho }_{N,\mu }\left(x\right):x\in K<\infty$ we have $sup{\varrho }_{M,\mu }\left(y\right):y\in {N}_{F}\left(K\right)<\infty$.

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