Extending Whitney’s extension theorem: nonlinear function spaces

Roberts, David Michael; Schmeding, Alexander

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.

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

[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 ${\mathrm{Lip}}_{\gamma }$-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)