Conditions on composition operators which map a space of Triebel-Lizorkin type into a Sobolev space. The case $1 < s < n/p$. II.

Preprint series: 98-30, Analysis

The paper is published: Forum Math., 10 (2), 199-231, 1998

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 $G : {Bbb R} \to {Bbb R}$ be a continuous function. Denote by $T_G$ the corresponding composition operator which sends $f$ to $G(f)$. Then we investigate necessary and sufficient conditions on the parameters $s, p,q,r$ and on the function $G$ such that an inclusion like
$T_G \, (F^s_{p,q}({Bbb R})) \subset {W}^m_{p} ({Bbb R})$
is true. Here $F^s_{p,q}$ denotes a space of Triebel-Lizorkin type and $W^m_p$ denotes a Sobolev space, respectively.
Necessary and sufficient conditions for such an inclusion to hold will be given in cases $G(t)= t^k, \: k \in {Bbb N}$, $G(t)= |t|^\mu$, $G(t)= t \, |t|^{\mu -1}, \: \mu >1$, $G \in C_0^\infty$, and $G$ a periodic $C^\infty$-function.

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