Regularity of the symbolic calculus in Besov algebras

Volume 184 / 2008

Gérard Bourdaud, Massimo Lanza de Cristoforis Studia Mathematica 184 (2008), 271-298 MSC: 46E35, 47H30. DOI: 10.4064/sm184-3-6


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.


  • Gérard BourdaudInstitut de Mathématiques de Jussieu
    Projet d'analyse fonctionnelle
    Case 186, 4 place Jussieu
    75252 Paris Cedex 05, France
  • Massimo Lanza de CristoforisDipartimento di Matematica Pura
    ed Applicata
    Università di Padova
    Via Trieste 63
    35121 Padova, Italy

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