no. 3
Extending Whitney’s extension theorem: nonlinear function spaces
[Extension du théorème d’extension de Whitney : espaces de fonctions non linéaires]
Roberts, David Michael1 ; Schmeding, Alexander2
1 School of Mathematical Sciences The University of Adelaide North Terrace Adelaide SA 5005 (Australia)
2 Department of Mathematics University of Bergen P.O. Box 7803 5020 Bergen (Norway)
Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1241-1286.

Nous considérons une version du problème de l’extension de Whitney, globale et non linéaire, pour les fonctions lisses à valeurs dans des variétés et définies sur des domaines fermés C à bords non-lisses dans des variétés possiblement non compactes. Supposant que C est une sous-variété à bord anguleux, ou qu’elle est compacte et localement convexe à bords non-lisses, nous montrons que l’opérateur de restriction, à partir des fonctions définies partout, est une submersion de variétés localement convexes, et donc possède des scindages linéaires locaux sur les cartes. Nous considérons à cet effet l’opérateur de restriction correspondant pour les espaces localement convexes de sections de fibrés vectoriels à support compact, permettant aussi de tariter le cas plus général où C n’a que des restrictions légères sur les cusps vers l’intérieur et l’extérieur, et montrons l’existence d’un opérateur de prolongement.

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains C, with non-smooth boundary, in possibly non-compact manifolds. Assuming C is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where C only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator.

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DOI : 10.5802/aif.3424
Classification : 58D15, 46T10, 58C07, 54C35, 46A04, 46A13, 53C21
Keywords: Whitney extension theorem, smooth functions on closed domain, Whitney jet, polynomial cusps, Fréchet space, submersion, manifolds with corners, manifolds with rough boundary, manifold of mappings, exponential law
Mot clés : théorème de l’extension de Whitney, fonctions lisses sur des domaines fermés, jet de Whitney, cuspides polynomiales, espace de Fréchet, submersion, variétés à bord anguleux, variétés à bords non-lisses, variétés d’applications, loi de l’exponentielle
Roberts, David Michael 1 ; Schmeding, Alexander 2

1 School of Mathematical Sciences The University of Adelaide North Terrace Adelaide SA 5005 (Australia)
2 Department of Mathematics University of Bergen P.O. Box 7803 5020 Bergen (Norway)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Roberts, David Michael; Schmeding, Alexander. Extending Whitney’s extension theorem:  nonlinear function spaces. Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1241-1286. doi : 10.5802/aif.3424. https://aif.centre-mersenne.org/articles/10.5802/aif.3424/

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