Superposition operator problems of Hölder-Lipschitz spaces

Abstract

Let

be a function defined on the real line, and

T f {T}_{f}

be the corresponding superposition operator which maps

h h

to

T f ( h ) {T}_{f}\left(h)

, i.e.,

T f ( h ) = f ∘ h {T}_{f}\left(h)=f\circ h

. In this article, the sufficient and necessary conditions such that

T f {T}_{f}

maps periodic Hölder-Lipschitz spaces

H p α {H}_{p}^{\alpha }

into itself with

0 < α < 1 p 0\lt \alpha \lt \frac{1}{p}

and

1 p < 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 < 1 p 0\lt \alpha \lt \frac{1}{p}

may be the first result about the superposition operator problems of smooth function space containing unbounded functions.