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Functional Calculus in Hölder-Zygmund Spaces

Authors: G. Bourdaud and Massimo Lanza de Cristoforis
Journal: Trans. Amer. Math. Soc. 354 (2002), 4109-4129
MSC (2000): Primary 46E35, 47H30
Published electronically: June 4, 2002
MathSciNet review: 1926867
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Abstract: In this paper we characterize those functions $f$ of the real line to itself such that the nonlinear superposition operator $T_{f}$ defined by $T_{f}[ g]:= f\circ g$ maps the Hölder-Zygmund space ${\mathcal C}^{s}({\mathbf R}^{n})$ to itself, is continuous, and is $r$ times continuously differentiable. Our characterizations cover all cases in which $s$ is real and $s>0$, and seem to be novel when $s>0$ is an integer.

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Additional Information

G. Bourdaud
Affiliation: Institut de Mathématiques de Jussieu, Équipe d’Analyse Fonctionnelle, Case 186, 4 place Jussieu, 75252 Paris Cedex 05, France

Massimo Lanza de Cristoforis
Affiliation: University of Padova, Dipartimento di Matematica Pura ed Applicata, Via Belzoni 7, 35131 Padova, Italia

Keywords: H\"older-Zygmund spaces, composition operators
Received by editor(s): June 13, 2000
Received by editor(s) in revised form: December 21, 2001
Published electronically: June 4, 2002
Additional Notes: The authors thank Jean-Pierre Kahane and Winfried Sickel for their help in the preparation of this paper. Massimo Lanza de Cristoforis wishes to thank Gérard Bourdaud and the University of Paris VII for hospitality during the months of September 1998 and of February 2000.
Article copyright: © Copyright 2002 American Mathematical Society