J. Rákosník (ed.),
Function spaces, differential operators and nonlinear analysis.
Proceedings of the conference held in Paseky na Jizerou, September 3-9, 1995.
Mathematical Institute, Czech Academy of Sciences, and Prometheus Publishing House, Praha 1996
p. 159 - 182

Composition operators acting on Sobolev spaces of fractional order---a survey on sufficient and necessary conditions

Winfried Sickel

Mathematisches Institut, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität, Leutragraben 1, 07743 Jena, Germany sickel@minet.uni-jena.de

Abstract: Let $A^s_{p,q}$ denote either a Besov space $\bspq(\Bbb R^n)$ or a Lizorkin--Triebel space $F^s_{p,q}(\Bbb R^n)$. Let $G:\Bbb R^n \to \Bbb R^n$ be continuous. Then the composition operator $T_G:f \to G(f)$ makes sense. It is known that one has the implication $$ T_G (A^s_{p,q}(\Bbb R^n)) \subset A^s_{p,q}(\Bbb R^n) \qquad \Longrightarrow \qquad G(t)=c \, t \quad \text{for some } c \in \Bbb R^n $$ provided that $1 \le p < \infty, \quad 1 \le q \le \infty \quad \text{and} \quad 1 + \frac{1}{p} < s < \frac{n}{p}$. In view of this negative result it is natural to ask about properties of the function $G$ implied by an embedding like $$ T_G (A^s_{p,q}\,(\Bbb R^n)) \subset B^r_{p,q} (\Bbb R^n) \tag 1 $$ for some $r$, $0<r<s<n/p$.

\proclaim{Theorem} Let $1 \le p < \infty$, $1 \le q \le \infty$ and $0 < r < s < n/p $. Suppose further $G \in C(\Bbb R^n)$. Then {\rm(1)} implies $G(0)=0$, $G \in B^{r,\text{loc}}_{p,q}(\Bbb R^n)$, and $$ \int_1^\infty w^{-\gamma p-1} \| G(\cdot) \psi(\frac{\cdot}{w}) | \dot{\Cal B}^r_{p,p}(\Bbb R^n)\|^p dw < \infty $$ for all\quad $\gamma > \dfrac{\frac{n}{p} + \frac{1}{p}(\frac{n}{p}-s)-r (\frac{n}{p}-s+1)}{\frac{n}{p}-s}$.

Here $\psi$ denotes a smooth cut-off function supported around the origin, $m>r$ some natural number and $$ \| f|\dot{\Cal B}^r_{p,p}(\Bbb R^n)\|^p = \int_{-1}^1 |h|^{-rp - 1} \int_{-\infty}^\infty \left| \sum_{j=0}^{m} (-1)^j \binom mj f(y+(m-j)h)\right|^p dy dh. $$ \endproclaim

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