no. 3
Extending Whitney’s extension theorem: nonlinear function spaces
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, 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.

Received:
Revised:
Accepted:
Online First:
Published online:
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 
@article{AIF_2021__71_3_1241_0,
     author = {Roberts, David Michael and Schmeding, Alexander},
     title = {Extending {Whitney{\textquoteright}s} extension theorem:  nonlinear function spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {1241--1286},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {3},
     year = {2021},
     doi = {10.5802/aif.3424},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3424/}
}
TY  - JOUR
AU  - Roberts, David Michael
AU  - Schmeding, Alexander
TI  - Extending Whitney’s extension theorem:  nonlinear function spaces
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 1241
EP  - 1286
VL  - 71
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3424/
DO  - 10.5802/aif.3424
LA  - en
ID  - AIF_2021__71_3_1241_0
ER  - 
%0 Journal Article
%A Roberts, David Michael
%A Schmeding, Alexander
%T Extending Whitney’s extension theorem:  nonlinear function spaces
%J Annales de l'Institut Fourier
%D 2021
%P 1241-1286
%V 71
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3424/
%R 10.5802/aif.3424
%G en
%F AIF_2021__71_3_1241_0
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/

[1] Alzaareer, Hamza; Schmeding, Alexander Differentiable mappings on products with different degrees of differentiability in the two factors, Expo. Math., Volume 33 (2015) no. 2, pp. 184-222 | DOI | MR | Zbl

[2] Bangert, Victor Totally convex sets in complete Riemannian manifolds, J. Differ. Geom., Volume 16 (1981) no. 2, pp. 333-345 | MR | Zbl

[3] Bastiani, Andree Applications différentiables et variétés différentiables de dimension infinie, J. Anal. Math., Volume 13 (1964), pp. 1-114 | DOI | MR | Zbl

[4] Bierstone, Edward Differentiable functions, Bol. Soc. Bras. Mat., Volume 11 (1980) no. 2, pp. 139-189 | DOI | MR | Zbl

[5] Brudnyi, Alexander; Brudnyi, Yuri Methods of geometric analysis in extension and trace problems. Volume 1, Monographs in Mathematics, 102, Springer, 2012, xxiv+560 pages | DOI | MR

[6] Etter, Daniel O.; Griffin, John S. On the metrisability of bundle space, Proc. Am. Math. Soc., Volume 5 (1954), pp. 466-467 | DOI | Zbl

[7] Fefferman, Charles Whitney’s extension problem for C m , Ann. Math., Volume 164 (2006) no. 1, pp. 313-359 | DOI | MR | Zbl

[8] Fefferman, Charles C m extension by linear operators, Ann. Math., Volume 166 (2007) no. 3, pp. 779-835 | DOI | MR | Zbl

[9] Frerick, Leonhard Extension operators for spaces of infinite differentiable Whitney jets, J. Reine Angew. Math., Volume 602 (2007), pp. 123-154 | DOI | MR | Zbl

[10] Frerick, Leonhard Stein’s extension operator for sets with Lip γ -boundary, Analysis (München), Volume 27 (2007) no. 2-3, pp. 251-259 | DOI | MR | Zbl

[11] Glöckner, Helge Infinite-dimensional Lie groups without completeness restrictions, Geometry and analysis on finite- and infinite-dimensional Lie groups (Będlewo, 2000) (Banach Center Publications), Volume 55, Polish Acad. Sci., 2002, pp. 43-59 | DOI | MR | Zbl

[12] Glöckner, Helge Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups, J. Funct. Anal., Volume 194 (2002) no. 2, pp. 347-409 | DOI | MR | Zbl

[13] Glöckner, Helge Lie groups over non-discrete topological fields (2004) (arxiv.org/abs/math/0408008)

[14] Glöckner, Helge Fundamentals of submersions and immersions between infinite-dimensional manifolds (2016) (arxiv.org/abs/1502.05795)

[15] Glöckner, Helge; Neeb, Karl-Hermann Infinite-dimensional Lie groups (in preparation)

[16] Heinonen, Juha; Kilpeläinen, Tero; Martio, Olli Nonlinear potential theory of degenerate elliptic equations, Dover Publications, 2006, xii+404 pages (Unabridged republication of the 1993 original) | MR

[17] Hjelle, Eivind Otto; Schmeding, Alexander Strong topologies for spaces of smooth maps with infinite-dimensional target, Expo. Math., Volume 35 (2017) no. 1, pp. 13-53 | DOI | MR | Zbl

[18] Jerison, David S.; Kenig, Carlos E. Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. Math., Volume 46 (1982) no. 1, pp. 80-147 | DOI | Zbl

[19] Karcher, Hermann Schnittort und konvexe Mengen in vollständigen Riemannschen Mannigfaltigkeiten, Math. Ann., Volume 177 (1968), pp. 105-121 | DOI | MR | Zbl

[20] Keller, Hans Heinrich Differential calculus in locally convex spaces, Lecture Notes in Mathematics, 417, Springer, 1974, iii+143 pages | DOI | MR

[21] Lee, John M. Introduction to smooth manifolds, Graduate Texts in Mathematics, 218, Springer, 2013, xvi+708 pages | MR

[22] Martio, Olli; Sarvas, Jukka Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn., Math., Volume 4 (1979) no. 2, pp. 383-401 | DOI | MR | Zbl

[23] Meise, Reinhold; Vogt, Dietmar Introduction to functional analysis, Oxford Graduate Texts in Mathematics, 2, Clarendon Press, 1997, x+437 pages | MR

[24] Michor, Peter W. Manifolds of differentiable mappings, Shiva Mathematics Series, 3, Shiva Publishing Ltd., 1980 | MR

[25] Mourgoglou, Mihalis On Harmonic Measure and Rectifiability in Uniform Domains, J. Geom. Anal., Volume 29 (2019), pp. 1193-1205 | DOI | MR | Zbl

[26] Neeb, Karl-Hermann Towards a Lie theory of locally convex groups, Jpn. J. Math., Volume 1 (2006) no. 2, pp. 291-468 | DOI | MR | Zbl

[27] Roberts, David Michael; Vozzo, Raymond F. The smooth Hom-stack of an orbifold, 2016 MATRIX Annals (Wood, David; Gier, Jan de; Praeger, Cheryl; Tao, Terence, eds.) (MATRIX Book Series), Volume 1, Springer, 2018 | DOI | MR | Zbl

[28] Schmeding, Alexander The diffeomorphism group of a non-compact orbifold, Diss. Math., Volume 507 (2015), p. 179 | DOI | MR | Zbl

[29] Wengenroth, Jochen Derived functors in functional analysis, Lecture Notes in Mathematics, 1810, Springer, 2003, viii+134 pages | DOI | MR

[30] Whitney, Hassler Analytic extensions of differentiable functions defined in closed sets, Trans. Am. Math. Soc., Volume 36 (1934), pp. 63-89 | DOI | MR | Zbl

[31] Whitney, Hassler Differentiable functions defined in closed sets. I, Trans. Am. Math. Soc., Volume 36 (1934) no. 2, pp. 369-387 | DOI | MR | Zbl

[32] Wockel, Christoph Smooth extensions and spaces of smooth and holomorphic mappings, J. Geom. Symmetry Phys., Volume 5 (2006), pp. 118-126 | MR | Zbl

[33] Wockel, Christoph Infinite-dimensional and higher structures in differential geometry (2013) (www.math.uni-hamburg.de/home/wockel/teaching/higher_structures.html, Lecture notes for a course given at the University of Hamburg)

Cited by Sources: