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Zeitschrift für Analysis und ihre Anwendungen

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Volume 18, Issue 2, 1999, pp. 205–229
DOI: 10.4171/ZAA/878

Published online: 1999-06-30

On the Autonomous Nemytskij Operator in Hölder Spaces

Manfred Goebel[1] and F. Sachweh[2]

(1) Universität Halle-Wittenberg, Germany
(2) Ahaus, Germany

The paper is devoted to the autonomous Nemytskij operator (superposition operator) in Hölder spaces $H^{k+\alpha}[a,b], (k, \alpha) \in \mathbb Z_+ \times [0, 1]. We study acting, continuity, Lipschitz continuity, and Fréchet differentiability conditions. For $k = 0$, $\alpha \in [0, 1]$ and $k \in \mathbb N, \alpha = 1$ the respective conditions are both necessary and sufficient. For $k \in \mathbb N, \alpha \in (0,1)$ only the acting condition is both necessary and sufficient; the other investigated properties are characterized by necessary and sufficient conditions different from each other.

Keywords: Hölder spaces, Lipschitz spaces, Nermytskij operator, superposition operator, acting conditions, boundedness, continuity and Lipschitz continuity, Fréchet differentiability

Goebel Manfred, Sachweh F.: On the Autonomous Nemytskij Operator in Hölder Spaces. Z. Anal. Anwend. 18 (1999), 205-229. doi: 10.4171/ZAA/878