Extending Whitney’s extension theorem: nonlinear function spaces
Annales de l'Institut Fourier, Volume 71 (2021) no. 3, pp. 1241-1286.

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.

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.

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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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Extending {Whitney{\textquoteright}s} extension theorem:  nonlinear function spaces},
     journal = {Annales de l'Institut Fourier},
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Roberts, David Michael; Schmeding, Alexander. Extending Whitney’s extension theorem:  nonlinear function spaces. Annales de l'Institut Fourier, Volume 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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