Algebras of pseudodifferential operators on complete manifolds
Authors:
Bernd Ammann, Robert Lauter and Victor Nistor
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 80-87
MSC (2000):
Primary 58J40; Secondary 58H05, 65R20
DOI:
https://doi.org/10.1090/S1079-6762-03-00114-8
Published electronically:
September 15, 2003
MathSciNet review:
2029468
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In several influential works, Melrose has studied examples of non-compact manifolds $M_0$ whose large scale geometry is described by a Lie algebra of vector fields $\mathcal V \subset \Gamma (M;TM)$ on a compactification of $M_0$ to a manifold with corners $M$. The geometry of these manifolds—called “manifolds with a Lie structure at infinity”—was studied from an axiomatic point of view in a previous paper of ours. In this paper, we define and study an algebra $\Psi _{1,0,\mathcal V}^\infty (M_0)$ of pseudodifferential operators canonically associated to a manifold $M_0$ with a Lie structure at infinity $\mathcal V \subset \Gamma (M;TM)$. We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to the algebras that we introduce. In particular, the algebra $\Psi _{1,0,\mathcal V}^\infty (M_0)$ is a “microlocalization” of the algebra $\textrm {Diff}^{*}_{\mathcal V}(M)$ of differential operators with smooth coefficients on $M$ generated by $\mathcal V$ and $\mathcal {C}^\infty (M)$. This proves a conjecture of Melrose (see his ICM 90 proceedings paper).
-
aln1 B. Ammann, R. Lauter, and V. Nistor. On the Riemannian geometry of manifolds with a Lie structure at infinity. To appear in Int. J. Math. and Math. Sci.
aln2 B. Ammann, R. Lauter, and V. Nistor. Pseudodifferential operators on manifolds with a Lie structure at infinity. Preprint, December 2002.
alnv1 B. Ammann, R. Lauter, V. Nistor, and A. Vasy. Complex powers and non-compact manifolds. To appear in Commun. Partial Differential Equations.
CrainicFernandez M. Crainic and R. L. Fernandes. Integrability of Lie brackets. Ann. of Math. 157 (2003), 575–620.
emmhei C. Epstein, R. B. Melrose, and G. Mendoza. The Heisenberg algebra, index theory and homology. In preparation.
- C. L. Epstein, R. B. Melrose, and G. A. Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains, Acta Math. 167 (1991), no. 1-2, 1–106. MR 1111745, DOI https://doi.org/10.1007/BF02392446
- Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536
- Max Karoubi, Homologie cyclique et $K$-théorie, Astérisque 149 (1987), 147 (French, with English summary). MR 913964 zfr R. Lauter. Pseudodifferential analysis on conformally compact spaces. Mem. Amer. Math. Soc., 163, 2003.
- Robert Lauter and Sergiu Moroianu, Fredholm theory for degenerate pseudodifferential operators on manifolds with fibered boundaries, Comm. Partial Differential Equations 26 (2001), no. 1-2, 233–283. MR 1842432, DOI https://doi.org/10.1081/PDE-100001754
- N. P. Landsman, M. Pflaum, and M. Schlichenmaier (eds.), Quantization of singular symplectic quotients, Progress in Mathematics, vol. 198, Birkhäuser Verlag, Basel, 2001. MR 1938548
- John N. Mather, Stratifications and mappings, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 195–232. MR 0368064
- Rafe Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16 (1991), no. 10, 1615–1664. MR 1133743, DOI https://doi.org/10.1080/03605309108820815
- Rafe R. Mazzeo and Richard B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), no. 2, 260–310. MR 916753, DOI https://doi.org/10.1016/0022-1236%2887%2990097-8
- Rafe Mazzeo and Richard B. Melrose, Pseudodifferential operators on manifolds with fibred boundaries, Asian J. Math. 2 (1998), no. 4, 833–866. Mikio Sato: a great Japanese mathematician of the twentieth century. MR 1734130, DOI https://doi.org/10.4310/AJM.1998.v2.n4.a9
- Richard B. Melrose, Transformation of boundary problems, Acta Math. 147 (1981), no. 3-4, 149–236. MR 639039, DOI https://doi.org/10.1007/BF02392873
- Richard B. Melrose, Pseudodifferential operators, corners and singular limits, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 217–234. MR 1159214
- Richard B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, vol. 4, A K Peters, Ltd., Wellesley, MA, 1993. MR 1348401
- Richard B. Melrose, Geometric scattering theory, Stanford Lectures, Cambridge University Press, Cambridge, 1995. MR 1350074
- Richard B. Melrose, Fibrations, compactifications and algebras of pseudodifferential operators, Partial differential equations and mathematical physics (Copenhagen, 1995; Lund, 1995) Progr. Nonlinear Differential Equations Appl., vol. 21, Birkhäuser Boston, Boston, MA, 1996, pp. 246–261. MR 1380995, DOI https://doi.org/10.1007/978-1-4612-0775-7_16 melcor R. B. Melrose. Geometric optics and the bottom of the spectrum. In F. Colombini and N. Lerner, editors, Geometrical optics and related topics, volume 32 of Progress in nonlinear differential equations and their applications. Birkhäuser, Basel-Boston-Berlin, 1997. MelroseMendoza R. B. Melrose and G. Mendoza. Elliptic operators of totally characteristic type. MSRI Preprint 1983.
- Victor Nistor, Groupoids and the integration of Lie algebroids, J. Math. Soc. Japan 52 (2000), no. 4, 847–868. MR 1774632, DOI https://doi.org/10.2969/jmsj/05240847
- Victor Nistor, Alan Weinstein, and Ping Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117–152. MR 1687747, DOI https://doi.org/10.2140/pjm.1999.189.117
- Cesare Parenti, Operatori pseudo-differenziali in $R^{n}$ e applicazioni, Ann. Mat. Pura Appl. (4) 93 (1972), 359–389. MR 437917, DOI https://doi.org/10.1007/BF02412028
- Michael Demuth, Elmar Schrohe, and Bert-Wolfgang Schulze (eds.), Boundary value problems, Schrödinger operators, deformation quantization, Mathematical Topics, vol. 8, Akademie Verlag, Berlin, 1995. Advances in Partial Differential Equations. MR 1389010
- Bert-Wolfgang Schulze, Boundary value problems and singular pseudo-differential operators, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1998. MR 1631763
- M. A. Shubin, Spectral theory of elliptic operators on noncompact manifolds, Astérisque 207 (1992), 5, 35–108. Méthodes semi-classiques, Vol. 1 (Nantes, 1991). MR 1205177
- Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463 Taylor2 M. Taylor. Partial differential equations, volumes I–III of Applied Mathematical Sciences. Springer-Verlag, New York, 1995–1997. ; ; ;
- András Vasy, Propagation of singularities in many-body scattering, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 3, 313–402 (English, with English and French summaries). MR 1839579, DOI https://doi.org/10.1016/S0012-9593%2801%2901066-7
- Jared Wunsch, Propagation of singularities and growth for Schrödinger operators, Duke Math. J. 98 (1999), no. 1, 137–186. MR 1687567, DOI https://doi.org/10.1215/S0012-7094-99-09804-6
Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (2000): 58J40, 58H05, 65R20
Retrieve articles in all journals with MSC (2000): 58J40, 58H05, 65R20
Additional Information
Bernd Ammann
Affiliation:
Universität Hamburg, Fachbereich 11–Mathematik, Bundesstrasse 55, D-20146 Hamburg, Germany
Email:
ammann@berndammann.de
Robert Lauter
Affiliation:
Universität Mainz, Fachbereich 17–Mathematik, D-55099 Mainz, Germany
Email:
lauter@mathematik.uni-mainz.de, lauterr@web.de
Victor Nistor
Affiliation:
Mathematics Department, Pennsylvania State University, University Park, PA 16802
Email:
nistor@math.psu.edu
Keywords:
Differential operator,
pseudodifferential operator,
principal symbol,
conormal distribution,
Riemannian manifold,
Lie algebra,
exponential map
Received by editor(s):
April 24, 2003
Published electronically:
September 15, 2003
Additional Notes:
Ammann was partially supported by the European Contract Human Potential Program, Research Training Networks HPRN-CT-2000-00101 and HPRN-CT-1999-00118; Nistor was partially supported by NSF Grants DMS 99-1981 and DMS 02-00808. Manuscripts available from http://www.math.psu.edu/nistor/.
Communicated by:
Michael E. Taylor
Article copyright:
© Copyright 2003
American Mathematical Society