Regularity of the symbolic calculus in Besov algebras
Volume 184 / 2008
Studia Mathematica 184 (2008), 271-298
MSC: 46E35, 47H30.
DOI: 10.4064/sm184-3-6
Abstract
We consider Besov and Lizorkin–Triebel algebras, that is, the real-valued function spaces $B_{{p},{q}}^{s}({\mathbb R}^n) \cap L_\infty(\mathbb R)$ and ${F_{{p},{q}}^{s}({\mathbb R}^n)} \cap L_\infty(\mathbb R)$ for all $s>0$. To each function $f:{\mathbb{R}}\to \mathbb{R} $ one can associate the composition operator $T_{f}$ which takes a real-valued function $g$ to the composite function $f\circ g$. We give necessary conditions and sufficient conditions on $f$ for the continuity, local Lipschitz continuity, and differentiability of any order of $T_{f}$ as a map acting in Besov and Lizorkin–Triebel algebras. In some cases, such as for $n=1$, such conditions turn out to be necessary and sufficient.
