On the Autonomous Nemytskij Operator in Hölder Spaces

M. Goebel and F. Sachweh

Abstract: The paper is devoted to the autonomous Nemytskij operator (superposition operator) in Hölder spaces $H^{k+\alpha}[a,b]$, $(k,\alpha) \in \Bbb{Z}_+ \times (0,1]$. We study acting, continuity, Lipschitz continuity, and Fréchet differentiability conditions. For $k = 0$, $\alpha \in (0,1]$ and $k \in \Bbb{N}$, $\alpha = 1$ the respective conditions are both necessary and sufficient. For $k \in \Bbb{N}$, $\alpha \in (0,1)$ 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, Nemytskij operator, superposition operator, acting conditions, boundedness, continuity and Lipschitz continuity, Fréchet differentiability

Full text of the article:

• Compressed DVI file (45 kilobytes)
• Compressed PostScript file (135 kilobytes)
• PDF file (289 kilobytes)