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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Superposition in homogeneous and vector valued Sobolev spaces


Author: Gérard Bourdaud
Journal: Trans. Amer. Math. Soc. 362 (2010), 6105-6130
MSC (2000): Primary 46E35, 47H30
Published electronically: June 16, 2010
MathSciNet review: 2661510
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Abstract: We give a sufficient condition on a function $ f:\mathbb{R}^{k}\rightarrow \mathbb{R}$ so that it takes by superposition the homogeneous vector valued space $ \dot{W}^{m}_{p}\cap \dot{W}^{1}_{mp}(\mathbb{R}^n, \mathbb{R}^k)$ into the corresponding real valued space, for integers $ m,n,k$ such that $ m\geq 2$, $ k,n\geq 1$, and $ p\in [1,+\infty[$. In case $ k=1$, this condition also turns out to be necessary. For $ k>1$, it is not proved to be necessary, but it is weaker than the conditions used till now, such as the continuity and boundedness of all derivatives up to order $ m$.


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Additional Information

Gérard Bourdaud
Affiliation: Institut de Mathématiques de Jussieu, Équipe d’Analyse Fonctionnelle, Université Paris Diderot, 175 rue du Chevaleret, 75013 Paris, France
Email: bourdaud@math.jussieu.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05150-5
PII: S 0002-9947(2010)05150-5
Keywords: Superposition operators, Sobolev spaces
Received by editor(s): August 26, 2008
Received by editor(s) in revised form: June 25, 2009
Published electronically: June 16, 2010
Article copyright: © Copyright 2010 American Mathematical Society