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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 193, Issue 5, pp 1519–1554 | Cite as

Composition operators acting on Besov spaces on the real line

  • Gérard BourdaudEmail author
  • Madani Moussai
  • Winfried Sickel
Article

Abstract

We study the composition operator \(T_f(g):= f\circ g\) on Besov spaces \(B_{{p},{q}}^{s}(\mathbb{R })\). In case \(1 < p< +\infty ,\, 0< q \le +\infty \) and \(s>1+ (1/p)\), we will prove that the operator \(T_f\) maps \(B_{{p},{q}}^{s}(\mathbb{R })\) to itself if, and only if, \(f(0)=0\) and \(f\) belongs locally to \(B_{{p},{q}}^{s}(\mathbb{R })\). For the case \(p=q\), i.e., in case of Slobodeckij spaces, we can extend our results from the real line to \(\mathbb{R }^n\).

Keywords

Homogeneous and inhomogeneous Besov spaces on the real line Slobodeckij spaces on \(\mathbb{R }^n\) Functions of bounded \(p\)-variation Composition operators Optimal inequalities 

Mathematics Subject Classification (2000)

46E35 47H30 

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Gérard Bourdaud
    • 1
    Email author
  • Madani Moussai
    • 2
  • Winfried Sickel
    • 3
  1. 1.IMJ-PRG (UMR 7586)Université Paris DiderotParis Cedex 13France
  2. 2.Department of Mathematics, LAFGEUniversity of M’SilaM’SilaAlgeria
  3. 3.Institute of MathematicsFriedrich-Schiller-University JenaJenaGermany

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