Composition operators on Besov algebras

Abstract

We study the composition operator $T_f(g):=f\circ g$ on Besov spaces $B_{p,q}^s(\mathbb{R})$ . In the case $1<p<+\infty$, $p\le q\le +\infty$ and $s>1+(1/p)$ we will prove that the operator $T_f$ takes $B_{p,q}^s(\mathbb{R})$ into itself if and only if $f(0)=0$ and $f$ belongs locally to $B_{p,q}^s(\mathbb{R})$ .