Regularity of the symbolic calculus in Besov algebras

Gérard Bourdaud ¹   ; Massimo Lanza de Cristoforis ²

¹ Institut de Mathématiques de Jussieu Projet d'analyse fonctionnelle Case 186, 4 place Jussieu 75252 Paris Cedex 05, France
² Dipartimento di Matematica Pura ed Applicata Università di Padova Via Trieste 63 35121 Padova, Italy

Studia Mathematica, Tome 184 (2008) no. 3, pp. 271-298

Résumé

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.

Détail

DOI : 10.4064/sm184-3-6

Keywords: consider besov lizorkin triebel algebras real valued function spaces mathbb cap infty mathbb mathbb cap infty mathbb each function mathbb mathbb associate composition operator which takes real valued function composite function circ necessary conditions sufficient conditions continuity local lipschitz continuity differentiability order map acting besov lizorkin triebel algebras cases conditions turn out necessary sufficient