The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (1379 KB) | Metadata | Table of Contents | ZAA summary
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