by **
W. Sickel**

**Preprint series:**
97-07, Analysis

**The paper is published:**
Forum Math., 9, 267-302, 1997

**MSC:**- 46E35 Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
- 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiui, Uryson, hysteresis operators, etc.), See also {45P05}

**Abstract:** Let $G :{Bbb R}^{n} \to {Bbb R}^{n}$ be a sufficiently smooth function. Denote by $T_G$ the corresponding composition operator which sends $f$ to $G(f)$.

Then we prove necessary conditions on $s,p,r,$ and $t$ such that the inclusion

\[ T_G \, ({\Bbb A}^{s}_{p,q}) \subset {\Bbb B}^r_{t, \infty} \]

holds. Here ${\Bbb A}^{s}_{p,q}$ stands for either a space of Triebel-Lizorkin type ${\Bbb F}^{s}_{p,q}$ or a space of Besov type ${\Bbb B}^{s}_{p,q}$.

**Keywords:** *composition operator, Nemytskij operator, Besov-Lizorkin-Triebel spaces*

**Upload:** 1999-02-10

**Update:** 1999-04-07

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