A necessary condition for composition in Besov spaces

ABSTRACT

In case $1<s<1+\left(1/p\right)$ we give a new necessary condition on a function $f:\mathbb{R}\to \mathbb{R}$, s.t. the associated composition operator $T_f\left(g\right):=f\circ g$ maps the function space $(B_{p,q}^s\cap L_{\mathrm{\infty }})\left({\mathbb{R}}^n\right)$ to itself. This condition is stronger than the classical one, i.e. f is locally Lipschitz continuous and belongs locally to $B_{p,q}^s\left(\mathbb{R}\right)$.

COMMUNICATED BY:

• M. Dalla Riva

KEYWORDS:

• Besov spaces
• Morrey spaces
• Besov-type spaces
• composition operators

AMS SUBJECT CLASSIFICATIONS:

• 46E35
• 47H30