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

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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 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 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**, DOI 10.1007/978-3-0348-8364-1
- 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 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 10.1016/0022-1236(87)90097-8
- 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 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 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**, DOI 10.1016/0377-0257(93)80040-i
- 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 10.1007/978-1-4612-0775-7_{1}6
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 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 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 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**, DOI 10.1515/9781400886104
- Nelson Dunford,
*A mean ergodic theorem*, Duke Math. J. **5** (1939), 635–646. MR **98**
- 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 10.1016/S0012-9593(01)01066-7
- Jared Wunsch,
*Propagation of singularities and growth for Schrödinger operators*, Duke Math. J. **98** (1999), no. 1, 137–186. MR **1687567**, DOI 10.1215/S0012-7094-99-09804-6

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.
emm91 C. Epstein, R. B. Melrose, and G. Mendoza. Resolvent of the Laplacian on strictly pseudoconvex domains. *Acta Math.* **167** (1991), 1–106.
hor3 L. Hörmander. *The analysis of linear partial differential operators, vol. 3. Pseudo-differential operators*, volume 274 of *Grundlehren der Mathematischen Wissenschaften*. Springer-Verlag, Berlin-Heidelberg-New York, 1985.
Karoubi M. Karoubi. Homologie cyclique et K-theorie. *Astérisque* **149** (1987), 1–147.
zfr R. Lauter. Pseudodifferential analysis on conformally compact spaces. *Mem. Amer. Math. Soc.*, 163, 2003.
defr R. Lauter and S. Moroianu. Fredholm theory for degenerate pseudodifferential operators on manifolds with fibered boundaries. *Commun. Partial Differential Equations* **26** (2001), 233–283.
LN1 R. Lauter and V. Nistor. Analysis of geometric operators on open manifolds: a groupoid approach. In N. P. Landsman, M. Pflaum, and M. Schlichenmaier, editors, *Quantization of Singular Symplectic Quotients*, volume 198 of *Progress in Mathematics*, pages 181–229. Birkhäuser, Basel-Boston-Berlin, 2001.
mather J. N. Mather. Stratifications and mappings. In *Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971)*, pages 195–232. Academic Press, New York, 1973.
Mazzeo R. Mazzeo. Elliptic theory of differential edge operators. I. *Commun. Partial Differ. Equations* **16** (1991), 1615–1664.
mame87 R. Mazzeo and R. B. Melrose. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. *J. Funct. Anal.* **75** (1987), 260–310.
MaMeAsian R. Mazzeo and R. B. Melrose. Pseudodifferential operators on manifolds with fibred boundaries. *Asian J. Math.* **2** (1998), 833–866.
me81 R. B. Melrose. Transformation of boundary value problems. *Acta Math.* **147** (1981), 149–236.
meicm R. B. Melrose. Pseudodifferential operators, corners and singular limits. In *Proceeding of the International Congress of Mathematicians, Kyoto*, pages 217–234, Berlin-Heidelberg-New York, 1990. Springer-Verlag.
meaps R. B. Melrose. *The Atiyah-Patodi-Singer index theorem.* Research Notes in Mathematics (Boston, Mass.). 4. Wellesley, MA: A. K. Peters, Ltd., 1993.
MelroseScattering R. B. Melrose. *Geometric scattering theory*. Stanford Lectures. Cambridge University Press, Cambridge, 1995.
mecom R. B. Melrose. Fibrations, compactifications and algebras of pseudodifferential operators. In L. Hörmander and A. Mellin, editors, *Partial Differential Equations and Mathematical Physics*, pages 246–261, 1996.
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.
NistorINT V. Nistor. Groupoids and the integration of Lie algebroids. *J. Math. Soc. Japan* **52** (2000), 847–868.
nwx V. Nistor, A. Weinstein, and P. Xu. Pseudodifferential operators on groupoids. *Pacific J. Math.* **189** (1999), 117–152.
Parenti C. Parenti. Operatori pseudodifferentiali in $\mathbb {R}^n$ e applicazioni. *Ann. Mat. Pura Appl.* **93** (1972), 359–389.
ScSc E. Schrohe and B.-W. Schulze. Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities. II. In *Boundary value problems, Schrödinger operators, deformation quantization*, volume 8 of *Math. Top.*, pages 70–205, Akademie Verlag, Berlin, 1995.
schwil B.-W. Schulze. *Boundary value problems and singular pseudo-differential operators.* Wiley-Interscience Series in Pure and Applied Mathematics. Chichester: John Wiley & Sons., 1998.
Shubin M. A. Shubin. Spectral theory of elliptic operators on noncompact manifolds. *Astérisque*, 207:5, 35–108, 1992. Méthodes semi-classiques, Vol. 1 (Nantes, 1991).
Taylor1 M. Taylor. *Pseudodifferential operators*, volume 34 of *Princeton Mathematical Series*. Princeton University Press, Princeton, NJ, 1981.
Taylor2 M. Taylor. *Partial differential equations*, volumes I–III of *Applied Mathematical Sciences*. Springer-Verlag, New York, 1995–1997. ; ; ;
VasyN A. Vasy. Propagation of singularities in many-body scattering. *Ann. Sci. École Norm. Sup. (4)* **34** (2001), 313–402.
jaredduke J. Wunsch. Propagation of singularities and growth for Schrödinger operators. *Duke Math. J.* **98** (1999), 137–186.

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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

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