Necessary conditions on composition operators acting between Besov spaces. The case $1 < s < n/p$. III.

Preprint series: 98-31, Analysis

The paper is published: Forum Math., 10 (3), 303-327, 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 consequences for the parameters $s, p,q$ and $r$ of the inclusion
$T_G \, (B^s_{p,q}({Bbb R})) \subset {B}^r_{p, \infty}({Bbb R}) .$
Here $B^{s}_{p,q}$ denotes a Besov space.

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