Pseudodifferential operators on manifolds with fibred corners

Debord, Claire; Lescure, Jean-Marie; Rochon, Frédéric

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1799-1880.

Un moyen géométrique d’encoder les singularités d’une pseudovariété stratifiée est de munir son intérieur d’une métrique cuspidale fibrée itérée. Pour une telle métrique, nous développons et étudions un calcul pseudodifférentiel généralisant le $\Phi$-calcul de Mazzeo et Melrose. Notre point de départ est l’observation bien connue qu’une pseudovariété stratifiée peut être « désingularisée » en variété à coins fibrés. Cela nous permet de définir les opérateurs pseudodifférentiels comme des distributions conormales sur un espace double éclaté approprié. Des applications symboles sont introduites, conduisant à la notion d’ellipticité pleine. Nous utilisons cela pour construire des paramétrix fins et pour caractériser les propriétés de nos opérateurs pseudodifférentiels, comme le fait d’être de Fredholm ou compacts. Nous introduisons aussi une version semi-classique du calcul que nous utilisons pour établir une dualité de Poincaré entre la $K$-homologie de la pseudovariété stratifiée et le $K$-groupe des opérateurs pleinement elliptiques.

One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the $\Phi$-calculus of Mazzeo and Melrose. Our starting point is the well-known observation that a stratified pseudomanifold can be ‘resolved’ into a manifold with fibred corners. This allows us to define pseudodifferential operators as conormal distributions on a suitably blown-up double space. Various symbol maps are introduced, leading to the notion of full ellipticity. This is used to construct refined parametrices and to provide criteria for the mapping properties of operators such as Fredholmness or compactness. We also introduce a semiclassical version of the calculus and use it to establish a Poincaré duality between the $K$-homology of the stratified pseudomanifold and the $K$-group of fully elliptic operators.

[1] Albin, Pierre; Leichtnam, Éric; Mazzeo, Rafe; Piazza, Paolo The signature package on Witt spaces, Ann. Sci. Éc. Norm. Supér. (4), Volume 45 (2012) no. 2, pp. 241-310 | Numdam | MR | Zbl

[2] Ammann, B.; Lauter, R.; Nistor, V. Pseudodifferential operators on manifolds with Lie structure at infinity, Annals of Mathematics, Volume 165 (2007), pp. 717-747 | DOI | MR | Zbl

[3] Anantharaman-Delaroche, C.; Renault, J. Amenable groupoids, L’Enseignement Mathématique, 2000 (Volume 36 of Monographies de L’Enseignement Mathématique) | MR | Zbl

[4] Androulidakis, Iakovos; Skandalis, Georges The holonomy groupoid of a singular foliation, J. Reine Angew. Math., Volume 626 (2009), pp. 1-37 | DOI | MR | Zbl

[5] Androulidakis, Iakovos; Skandalis, Georges The analytic index of elliptic pseudodifferential operators on a singular foliation, J. K-Theory, Volume 8 (2011) no. 3, pp. 363-385 | DOI | MR | Zbl

[6] Bacuta, Constantin; Mazzucato, Anna L.; Nistor, Victor; Zikatanov, Ludmil Interface and mixed boundary value problems on $n$-dimensional polyhedral domains, Doc. Math., Volume 15 (2010), pp. 687-745 | MR | Zbl

[7] Blackadar, Bruce $K$-theory for operator algebras, Cambridge University Press, 1998 | MR | Zbl

[8] Brasselet, J.-P.; Hector, G.; Saralegi, M. Théorème de de Rham pour les variétés stratifiées, Ann. Global Anal. Geom., Volume 9 (1991) no. 3, pp. 211-243 | DOI | MR | Zbl

[9] Cheeger, J. Spectral geometry of singular Riemannian spaces, J. Differential Geom., Volume 18 (1983) no. 4, pp. 575-657 | MR | Zbl

[10] Connes, A. Noncommutative Geometry, Academic Press, San Diego, CA, 1994 | MR | Zbl

[11] Debord, Claire Holonomy groupoids of singular foliations, J. Differential Geom., Volume 58 (2001) no. 3, pp. 467-500 | MR | Zbl

[12] Debord, Claire; Lescure, Jean-Marie $K$-duality for stratified pseudomanifolds, Geom. Topol., Volume 13 (2009) no. 1, pp. 49-86 | DOI | MR | Zbl

[13] Debord, Claire; Lescure, Jean-Marie Index theory and groupoids, Geometric and topological methods for quantum field theory, Cambridge Univ. Press (2010), pp. 86-158 | MR | Zbl

[14] Dixmier, J. Les $C^{*}$-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964, pp. xi+382 | MR | Zbl

[15] Epstein, C.L.; Mendoza, G.A.; Melrose, R.B. Resolvent of the Laplacian on strictly pseudoconvex domains., Acta Math., Volume 167 (1991), pp. 1-106 | DOI | MR | Zbl

[16] Gil, Juan B.; Krainer, Thomas; Mendoza, Gerardo A. Dynamics on Grassmannians and resolvents of cone operators, Anal. PDE, Volume 4 (2011) no. 1, pp. 115-148 | DOI | MR | Zbl

[17] Grieser, D.; Hunsicker, E. Pseudodifferential calculus for generalized $\mathbb{Q}$-rank 1 locally symmetric spaces I, J. Funct. Anal., Volume 257 (2009) no. 12, pp. 3748-3801 | DOI | MR | Zbl

[18] Grusin, V. V. A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold, Mat. Sb. (N.S.), Volume 84 (126) (1971), pp. 163-195 | MR

[19] Hörmander, L. The Analysis of Linear Partial Differential Operators. Vol. 3, Springer-Verlag, Berlin, 1985 | Zbl

[20] Jeffres, T.; Mazzeo, R.; Rubinstein, Y. Kähler-Einstein metrics with edge singularities (http://arxiv.org/abs/1105.5216)

[21] Karoubi, M. K-Theory, Springer-Verlag, 2008 (reprint of the 1978 edition) | MR | Zbl

[22] Kasparov, G.G. Equivariant KK-theory and the Novikov conjecture, Invent. Math., Volume 91 (1988), pp. 147-201 | DOI | MR | Zbl

[23] Krainer, T. Elliptic boundary problems on manifolds with polycylindrical ends, J. Funct. Anal., Volume 244 (2007), pp. 351-386 | DOI | MR | Zbl

[24] Lauter, R.; Monthubert, B.; Nistor, V. Pseudodifferential Analysis on Continuous Family Groupoids, Documenta Math., Volume 5 (2000), pp. 625-655 | MR | Zbl

[25] Lauter, Robert; Monthubert, Bertrand; Nistor, Victor Spectral invariance for certain algebras of pseudodifferential operators, J. Inst. Math. Jussieu, Volume 4 (2005) no. 3, pp. 405-442 | DOI | MR | Zbl

[26] Mackenzie, Kirill C. H. General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005, pp. xxxviii+501 | MR | Zbl

[27] Mazya, V.; Plamenevskii, B.A. Elliptic boundary value problems on manifolds with singularities, Probl. Mat. Anal., Volume 6 (1977), pp. 85-142 | Zbl

[28] Mazzeo, R. Elliptic theory of differential edge operators. I., Comm. Partial Differential Equations, Volume 16 (1991) no. 10, pp. 1615-1664 | DOI | MR | Zbl

[29] Mazzeo, R.; Melrose, R.B. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., Volume 75 (1987) no. 2, pp. 260-310 | DOI | MR | Zbl

[30] Mazzeo, R.; Melrose, R.B. Analytic surgery and the eta invariant, Geom. Funct. Anal., Volume 5 (1995) no. 1, pp. 14-75 | DOI | MR | Zbl

[31] Mazzeo, R.; Melrose, R.B. Pseudodifferential operators on manifolds with fibred boundaries, Asian J. Math., Volume 2 (1999) no. 4, pp. 833-866 | MR | Zbl

[32] Mazzeo, R.; Montcouquiol, G. Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra, J. Differential Geom., Volume 87 (2011) no. 3, pp. 525-576 | MR | Zbl

[33] Melrose, R.B. Differential analysis on manifolds with corners (http://www-math.mit.edu/~rbm/book.html)

[34] Melrose, R.B. Calculus of conormal distributions on manifolds with corners, Int. Math. Res. Notes, Volume 3 (1992), pp. 51-61 | DOI | MR | Zbl

[35] Melrose, R.B. The Atiyah-Patodi-Singer index theorem, A. K. Peters, Wellesley, Massachusetts, 1993 | MR | Zbl

[36] Melrose, R.B. The eta invariant and families of pseudodifferential operators, Math. Res. Lett., Volume 2 (1995) no. 5, pp. 541-561 | DOI | MR | Zbl

[37] Melrose, R.B. Geometric Scattering theory, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[38] Melrose, R.B.; Piazza, P. Analytic K-theory for manifolds with corners, Adv. in Math, Volume 92 (1992), pp. 1-27 | DOI | MR | Zbl

[39] Melrose, R.B.; Rochon, F. Index in K-theory for Families of Fibred Cusp Operators, K-theory, Volume 37 (2006), pp. 25-104 | DOI | MR | Zbl

[40] Monthubert, Bertrand; Pierrot, François Indice analytique et groupoïdes de Lie, C. R. Acad. Sci. Paris Sér. I Math., Volume 325 (1997), pp. 193-198 | DOI | MR | Zbl

[41] Muhly, Paul S.; Renault, Jean N.; Williams, Dana P. Equivalence and isomorphism for groupoid $C^{*}$-algebras, J. Operator Theory, Volume 17 (1987), pp. 3-22 | MR | Zbl

[42] Nazaikinskii, V.; Savin, A.; Sternin, B. Homotopy classification of elliptic operators on stratified manifolds, Izv. Math., Volume 71 (2007) no. 6, pp. 1167-1192 | DOI | MR | Zbl

[43] Nazaikinskii, V.; Savin, A.; Sternin, B. Pseudodifferential operators on stratified manifolds, Differ. Uravn., Volume 43 (2007) no. 4, pp. 519-532 | MR | Zbl

[44] Nazaikinskii, V.; Savin, A.; Sternin, B. Pseudodifferential operators on stratified manifolds II, Differ. Uravn., Volume 43 (2007) no. 5, pp. 685-696 | MR | Zbl

[45] Nazaikinskii, Vladimir; Savin, Anton; Sternin, Boris Elliptic theory on manifolds with corners. I. Dual manifolds and pseudodifferential operators, $C^{*}$-algebras and elliptic theory II (Trends Math.), Birkhäuser, Basel, 2008, pp. 183-206 | DOI | MR | Zbl

[46] Nazarov, Sergey A.; Plamenevskii, Boris A. Elliptic problems in domains with piecewise smooth boundaries, de Gruyter Expositions in Mathematics, 13, Walter de Gruyter & Co., Berlin, 1994, pp. viii+525 | DOI | MR | Zbl

[47] Nistor, V.; Weinstein, A.; Xu, P. Pseudodifferential operators on groupoids, Pacific J. Math., Volume 189 (1999), pp. 117-152 | DOI | MR | Zbl

[48] Parenti, C. Operatori pseudodifferenziali in ${ℝ}^n$ e applicazioni, Annali Mat. Pura et Appl., Volume 93 (1972), pp. 359-389 | DOI | MR | Zbl

[49] Paterson, Alan L. T Continuous family groupoids, Homology, Homotopy and Applications, Volume 2 (2000), pp. 89-104 | DOI | MR | Zbl

[50] Renault, Jean A groupoid approach to $C^{*}$-algebras, Lecture Notes in Mathematics, 793, Springer, Berlin, 1980, pp. ii+160 | MR | Zbl

[51] Rochon, Frédéric Pseudodifferential operators on manifolds with foliated boundaries, J. Funct. Anal., Volume 262 (2012) no. 3, pp. 1309-1362 | DOI | MR | Zbl

[52] Rochon, Frédéric; Zhang, Zhou Asymptotics of complete Kähler metrics of finite volume on quasiprojective manifolds, Adv. Math., Volume 231 (2012) no. 5, pp. 2892-2952 | DOI | MR | Zbl

[53] Savin, Anton Elliptic operators on manifolds with singularities and K-homology, K-theory, Volume 34 (2005), pp. 71-98 | DOI | MR | Zbl

[54] Schulze, B.-W. Pseudo-differential operators on manifolds with singularities, North-Holland, Amsterdam, 1991 | MR | Zbl

[55] Schulze, B.-W. Iterative Structures on Singular Manifolds, Geometric and Spectral Analysis (CRM proceedings), American mathematical society, 2014, pp. 173-222 | MR

[56] Shubin, M.A. Pseudodifferential operators on ${ℝ}^n$, Sov. Math. Dokl., Volume 12 (1971), pp. 147-151 | Zbl

[57] Shubin, M.A. Pseudodifferential operators and Spectral Theory, Springer, 2001 | MR | Zbl

[58] Skandalis, Georges Kasparov’s bivariant $K$-theory and applications, Exposition. Math., Volume 9 (1991), pp. 193-250 | MR | Zbl

[59] Treves, F. Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967 | MR | Zbl

[60] Vassout, Stéphane Unbounded pseudodifferential calculus on Lie groupoids, J. Funct. Anal., Volume 236 (2006) no. 1, pp. 161-200 | DOI | MR | Zbl

[61] Vasy, Andras Asymptotic behavior of generalized eigenfunctions in $N$-body scattering, J. Funct. Anal., Volume 148 (1997) no. 1, pp. 170-184 | DOI | MR | Zbl

[62] Verona, Andrei Stratified mappings—structure and triangulability, Lecture Notes in Mathematics, 1102, Springer-Verlag, Berlin, 1984, pp. ix+160 | MR | Zbl