Superposition operator problems of Hölder-Lipschitz spaces

Abstract

Let ff be a function defined on the real line, and Tf{T}_{f} be the corresponding superposition operator which maps hh to Tf(h){T}_{f}\left(h), i.e., Tf(h)=f∘h{T}_{f}\left(h)=f\circ h. In this article, the sufficient and necessary conditions such that Tf{T}_{f} maps periodic Hölder-Lipschitz spaces Hpα{H}_{p}^{\alpha } into itself with 0<α<1p0\lt \alpha \lt \frac{1}{p} and 1p<α<1\frac{1}{p}\lt \alpha \lt 1, where α\alpha is the smoothness index, are shown. Our result in the case 0<α<1p0\lt \alpha \lt \frac{1}{p} may be the first result about the superposition operator problems of smooth function space containing unbounded functions.

Keywords

• hölder-lipschitz spaces
• superposition operator problem
• locally lipschitz function
• lipschitz function
• 26a15
• 26a16
• 47h30