Necessary conditions on composition operators acting on Sobolev spaces of fractional order. The critical case $1 < s < n/p$.

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