On automatic boundedness of Nemytskiĭ set-valued operators

Rolewicz, S.; Song, Wen

Rolewicz, S. ; Song, Wen

Studia Mathematica, Tome 113 (1995), p. 65-72 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $F : Ω \times X \to 2^{Y}$ . It is shown that if $N_{F}$ maps a modular space $(N (L (Ω, Σ, μ; X)), ϱ_{N, μ})$ into subsets of a modular space $(M (L (Ω, Σ, μ; Y)), ϱ_{M, μ})$ , then $N_{F}$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_{K} = s u p ϱ_{N, μ} (x) : x \in K < \infty$ we have $s u p ϱ_{M, μ} (y) : y \in N_{F} (K) < \infty$ .

Publié le : 1995-01-01

Zbl 0826.47052

EUDML-ID : urn:eudml:doc:216160

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     author = {S. Rolewicz and Wen Song},
     title = {On automatic boundedness of Nemytski\u\i\ set-valued operators},
     journal = {Studia Mathematica},
     volume = {113},
     year = {1995},
     pages = {65-72},
     zbl = {0826.47052},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv113i1p65bwm}
}

Rolewicz, S.; Song, Wen. On automatic boundedness of Nemytskiĭ set-valued operators. Studia Mathematica, Tome 113 (1995) pp. 65-72. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv113i1p65bwm/

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