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 whose large scale geometry is described by a Lie algebra of vector fields on a *compactification* of to a manifold with corners . 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 of pseudodifferential operators canonically associated to a manifold with a Lie structure at infinity . 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 is a ``microlocalization'' of the algebra of differential operators with smooth coefficients on generated by and . This proves a conjecture of Melrose (see his ICM 90 proceedings paper).

**1.**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.**2.**B. Ammann, R. Lauter, and V. Nistor. Pseudodifferential operators on manifolds with a Lie structure at infinity. Preprint, December 2002.**3.**B. Ammann, R. Lauter, V. Nistor, and A. Vasy. Complex powers and non-compact manifolds. To appear in*Commun. Partial Differential Equations*.**4.**M. Crainic and R. L. Fernandes. Integrability of Lie brackets.*Ann. of Math.***157**(2003), 575-620.**5.**C. Epstein, R. B. Melrose, and G. Mendoza. The Heisenberg algebra, index theory and homology. In preparation.**6.**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**, https://doi.org/10.1007/BF02392446**7.**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****8.**Max Karoubi,*Homologie cyclique et 𝐾-théorie*, Astérisque**149**(1987), 147 (French, with English summary). MR**913964****9.**R. Lauter. Pseudodifferential analysis on conformally compact spaces.*Mem. Amer. Math. Soc.*, 163, 2003.**10.**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**, https://doi.org/10.1081/PDE-100001754**11.**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****12.**John N. Mather,*Stratifications and mappings*, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 195–232. MR**0368064****13.**Rafe Mazzeo,*Elliptic theory of differential edge operators. I*, Comm. Partial Differential Equations**16**(1991), no. 10, 1615–1664. MR**1133743**, https://doi.org/10.1080/03605309108820815**14.**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**, https://doi.org/10.1016/0022-1236(87)90097-8**15.**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**, https://doi.org/10.4310/AJM.1998.v2.n4.a9**16.**Richard B. Melrose,*Transformation of boundary problems*, Acta Math.**147**(1981), no. 3-4, 149–236. MR**639039**, https://doi.org/10.1007/BF02392873**17.**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****18.**Richard B. Melrose,*The Atiyah-Patodi-Singer index theorem*, Research Notes in Mathematics, vol. 4, A K Peters, Ltd., Wellesley, MA, 1993. MR**1348401****19.**Richard B. Melrose,*Geometric scattering theory*, Stanford Lectures, Cambridge University Press, Cambridge, 1995. MR**1350074****20.**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**, https://doi.org/10.1007/978-1-4612-0775-7_16**21.**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.**22.**R. B. Melrose and G. Mendoza. Elliptic operators of totally characteristic type. MSRI Preprint 1983.**23.**Victor Nistor,*Groupoids and the integration of Lie algebroids*, J. Math. Soc. Japan**52**(2000), no. 4, 847–868. MR**1774632**, https://doi.org/10.2969/jmsj/05240847**24.**Victor Nistor, Alan Weinstein, and Ping Xu,*Pseudodifferential operators on differential groupoids*, Pacific J. Math.**189**(1999), no. 1, 117–152. MR**1687747**, https://doi.org/10.2140/pjm.1999.189.117**25.**Cesare Parenti,*Operatori pseudo-differenziali in 𝑅ⁿ e applicazioni*, Ann. Mat. Pura Appl. (4)**93**(1972), 359–389. MR**437917**, https://doi.org/10.1007/BF02412028**26.**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****27.**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****28.**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****29.**Michael E. Taylor,*Pseudodifferential operators*, Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981. MR**618463****30.**Michael E. Taylor,*Partial differential equations*, Texts in Applied Mathematics, vol. 23, Springer-Verlag, New York, 1996. Basic theory. MR**1395147****31.**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**, https://doi.org/10.1016/S0012-9593(01)01066-7**32.**Jared Wunsch,*Propagation of singularities and growth for Schrödinger operators*, Duke Math. J.**98**(1999), no. 1, 137–186. MR**1687567**, https://doi.org/10.1215/S0012-7094-99-09804-6

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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

DOI:
https://doi.org/10.1090/S1079-6762-03-00114-8

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