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An Approach to a Version of the S(M, g)-pseudo-differential Calculus on Manifolds

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Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations

Part of the book series: Operator Theory: Advances and Applications ((APDE,volume 145))

Abstract

For appropriate triples (M,g,M), where M is an (in general non-compact) manifold, g is a metric on T* M, and M is a weight function on T* M, we develop a pseudo-differential calculus on.A4 which is based on the S(M,g))-calculus of L. Hörmander [27] in local models. In order to do so, we generalize the concept of E. Schrohe [41] of so-called SG-compatible manifolds. In the final section we give an outlook onto topological properties of the algebras of pseudo-differential operators. We state the existence of “order reducing operators” and that the algebra of operators of order zero is a submultiplicative Ψ*-algebra in the sense of B. Gramsch [18] in \( \mathcal{L}\left( {{L^2}\left( M \right)} \right) \).

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References

  1. F. Baldus. Weyl-Hörmander-Quantisierung auf dem \( {\mathbb{R}^n}\), Spektralinvarianz and Submultiplikativität der durch Kommutatoren definierten Fréchetalgebren. Diplomarbeit, Fachbereich 17-Mathematik, Johannes Gutenberg-Universität Mainz, 1996.

    Google Scholar 

  2. F. Baldus. S (M, g)-pseudo-differential calculus with spectral invariance on \( {\mathbb{R}^n}\) and manifolds for Banach function spaces. PhD thesis, Johannes Gutenberg-Universitat Mainz, 2001, Logos Verlag, Berlin, 2001. 300 pages.

    Google Scholar 

  3. R. Beals. A general calculus of pseudodifferential operators. Duke Math. J., 42:1–42, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Beals. Characterization of pseudodifferential operators and applications. Duke Math. J.,44:45–57, 1977. ibid. 46:215, 1979.

    Google Scholar 

  5. R. Beals. Weighted distribution spaces and pseudodifferential operators. J. An. Math., 39:130–187, 1981.

    MathSciNet  Google Scholar 

  6. R. Beals and C. Fefferman. Spatially inhomogeneous pseudodifferential operators I. Comm. Pure Appl. Math., 27:1–24, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Beals and P. Greiner. Calculus on Heisenberg manifolds, volume 119 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1988.

    Google Scholar 

  8. J.-M. Bony. Fourier integral operators and Weyl-Hormander calculus. In J.-M. Bony and M. Morimoto, editors, New trends in microlocal analysis, pages 3–22. SpringerVerlag, Tokyo - Berlin - Heidelberg - New York, 1997.

    Chapter  Google Scholar 

  9. J.-M. Bony and J.-Y. Chemin. Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bull. Soc. Math. France, 122:77–118, 1994.

    MathSciNet  MATH  Google Scholar 

  10. J.-M. Bony and N. Lerner. Quantification asymptotique et microlocalisations d’ordre supérieur I. Ann. Sci. Ecole Norm. Sup. 4 e s é rie, 22:377–433, 1989.

    MathSciNet  MATH  Google Scholar 

  11. H.O. Cordes. A global parametrix for pseudodifferential operators over \( {\mathbb{R}^n}\). Preprint No. 90, SFB 72, Bonn, 1976.

    Google Scholar 

  12. H.O. Cordes. On some C* -algebras and Frechet* -algebras of pseudodifferential operators, volume 43 of Proc. Symp. in Pure Math. - Pseudodifferential operators, pages 79–104. Amer. Math. Soc., Providence, Rhode Island, 1985.

    Google Scholar 

  13. H.O. Cordes. The technique of pseudodifferential operators, volume 202 of London Mathematical Society, Lecture Note Series. Cambridge University Press, Cambridge - London - New York, 1995.

    Book  Google Scholar 

  14. H.O. Cordes and S.H. Doong. The Laplace comparison algebra of spaces with conical and cylindrical ends. In H.O. Cordes, B. Gramsch, and H. Widom, editors, Pseudo-differential operators, volume 1256 of Lect. Notes Math., pages 55–90, Berlin - Heidelberg - New York - London - Paris - Tokyo, 1987. Proc. Conf., Oberwolfach/Ger. 1986, Springer-Verlag.

    Chapter  Google Scholar 

  15. N. Dencker. The Weyl calculus with locally temperate metrics and weights. Ark. Mat., 24:59–79, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  16. Y.V. Egorov and B.-W. Schulze. Pseudo-differential Operators, Singularities, Applications, volume 93 of Operator Theory, Advances and Applications. Birkhauser Verlag, Basel - Boston - Berlin, 1997.

    Google Scholar 

  17. K.O. Friedrichs. Pseudo-Differential Operators, An Introduction. Lecture Notes, Courant Inst. Math. Sci. New York University, 1968.

    Google Scholar 

  18. B. Gramsch. Some homogeneous spaces in the operator theory and Ψ-algebras. In Tagungsbericht Oberwolfach 42/81- Funktionalanalysis: C* -Algebren, 1981.

    Google Scholar 

  19. B. Gramsch. Relative Inversion in der Störungstheorie von Operatoren and Ψ -Algebren. Math. Annalen,269:27–71, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  20. B. Gramsch. Oka’s principle for special Frechet Lie groups and homogeneous manifolds in topological algebras of the microlocal analysis. In Banach algebras 97 Proceedings, pages 189–204, Berlin - New York, 1998. de Gruyter.

    Google Scholar 

  21. B. Gramsch and W. Kaballo. Multiplicative decompositions of holomorphic Fred-holm functions and Ψ*-algebras. Math. Nachr., 204:83–100, 1999.

    MathSciNet  MATH  Google Scholar 

  22. B. Gramsch, J. Ueberberg, and K. Wagner. Spectral invariance and submultiplicativity for Fréchet algebras with applications to pseudo-differential operators and Ψ*-quantization. In Operator theory: Advances and Applications, vol. 57, pages 71–98, Birkhäuser, Basel, 1992.

    Google Scholar 

  23. B. Helffer. Théorie spectrale pour des opérateurs globalement elliptiques, volume 112 of Asterisque. Société Mathematique de France, 1984.

    MATH  Google Scholar 

  24. L. Hörmander. Pseudo-differential operators. Comm. Pure Appl. Math., 18:501–517, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Hörmander. Pseudo-differential operators and hypoelliptic equations, volume X of Proc. Symp. in Pure Math. - Singular Integrals, pages 138–183. Amer. Math. Soc., Providence, Rhode Island, 1966.

    Google Scholar 

  26. L. Hörmander. Fourier integral operators I. Acta Math., 127:79–183, 1971.

    Article  MathSciNet  MATH  Google Scholar 

  27. L. Hörmander. The Weyl calculus of pseudo-differential operators. Comm. Pure Appl. Math., 32:359–443, 1979.

    Article  MATH  Google Scholar 

  28. L. Hörmander. The analysis of linear partial differential operators, vol. 3. SpringerVerlag, Berlin - Heidelberg - New York, 1985.

    Google Scholar 

  29. J.J. Kohn and L. Nirenberg. An algebra of pseudo-differential operators. Comm. Pure Appl. Math.,18:269–305, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  30. H. Kumano-go and K. Taniguchi. Oscillatory integrals of symbols of pseudo-differential operators on \( {\mathbb{R}^n}\) and operators of Fredholm type. Proc. Japan Acad., 49:397–402, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Lauter. Holomorphic functional calculus in several variables and Ψ*-algebras of totally characteristic operators on manifolds with boundary. PhD thesis, Johannes Gutenberg-Universität Mainz, 1996, Shaker Verlag, Aachen, 1997,241 pages.

    Google Scholar 

  32. R. Lauter and V. Nistor. Analysis of geometric operators on open manifolds: a groupoid approach. Heft 108, Universität Münster, SFB 478 Geometrische Strukturen in der Mathematik, May 2000.

    Google Scholar 

  33. R.B. Melrose. Spectral and scattering theory for the Laplacian on asymptotically Euclidean space. In M. Ikawa, editor, Spectral and Scattering Theory, volume 162 of Lecture Notes in Pure and Applied Mathematics, pages 85–130, New York, 1994. Marcel Dekker Inc. Proceedings of the Taniguchi International Workshop held in Sanda, November 1992.

    Google Scholar 

  34. R.B. Melrose. Geometric scattering theory. Cambridge University Press, 1995.

    MATH  Google Scholar 

  35. L. Nirenberg. Pseudo-differential operators. Amer. Math. Soc. Symp. Pure Math., 16:149–167, 1970.

    MathSciNet  Google Scholar 

  36. C. Parenti. Operatori pseudo-differenziali in \( {\mathbb{R}^n}\) e applicazioni. Ann. Mat. Pura Appl. (4), 93:359–389, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  37. B.A. Plamenevskij and V.N. Senichkin. Pseudodifferential operators on manifolds with singularities. Funct. Anal. Appl., 33(2):154–156, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  38. B.A. Plamenevskij and V.N. Senichkin. A class of pseudodifferential operators in \( {\mathbb{R}^m}\) and on stratified manifolds. Mat. Sb., 191(5):725–757, 2000.

    Article  MathSciNet  Google Scholar 

  39. Yu. Safarov. Pseudodifferential operators and linear connections. Proc. London Math. Soc., 74(3):379–416, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  40. E. Schrohe. Komplexe Potenzen elliptischer Pseudodifferentialoperatoren. PhD thesis, Johannes Gutenberg-Universität Mainz, 1986.

    Google Scholar 

  41. E. Schrohe. Spaces of weighted symbols and weighted Sobolev spaces on manifolds. In H.O. Cordes, B. Gramsch, and H. Widom, editors, Pseudo-differential operators,volume 1256 of Lect. Notes Math.,pages 360–377, Berlin - Heidelberg - New York -London - Paris - Tokyo, 1987. Proc. Conf., Oberwolfach/Ger. 1986, Springer-Verlag.

    Chapter  Google Scholar 

  42. E. Schrohe. A Ψ*-algebra of pseudo-differential operators on noncompact manifolds. Arch. Math., 51:81–86, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  43. E. Schrohe. Boundedness and spectral invariance for standard pseudo-differential operators on anisotropically weighted L p -Sobolev-spaces. Integral equations operator theory,13(2):271–284, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  44. E. Schrohe. Fréchet algebra techniques for boundary value problems on noncompact manifolds: Fredholm criteria and functional calculus via spectral invariance. Math. Nachrichten,199:145–185, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  45. M.A. Shubin. Pseudodifferential operators in \(\mathbb{R}^2 \). Soy. Math. Dokl., 12, Ser. N1, pages 147–151, 1971.

    Google Scholar 

  46. M.A. Shubin. Pseudodifferential Operators and Spectral Theory. Springer-Verlag, Berlin - Heidelberg - New York - London - Paris - Tokyo, 1987.

    Book  MATH  Google Scholar 

  47. M.A. Shubin. Spectral theory of elliptic operators on non-compact manifolds. In Methodes semi-classiques, Volume 1,École d’Éte, Nantes 1991, volume 207 of Astérisque, pages 35–108. Société Mathématique de France, 1992.

    Google Scholar 

  48. J. Ueberberg. Zur Spektralinvarianz von Algebren von Pseudo-Differential Operatoren in der L P-Theorie. Manuscripta Math., 61:459–475, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  49. A. Unterberger. Encore des classes de symboles. In Sém. Goulaouic-Schwartz 1977–1978, Ecole Polytechnique, Paris, 1977, 17 pages.

    Google Scholar 

  50. A. Unterberger. Oscillateur harmonique et operateurs pseudo-differentiels. Ann. Inst. Fourier, 29–3:201–221, 1979.

    Article  MathSciNet  Google Scholar 

  51. A. Unterberger and J. Bokobza. Les opérateurs de Calderon-Zygmund précisés. C. R. Acad. Sci. Paris, 259:1612–1614, 1964.

    MathSciNet  MATH  Google Scholar 

  52. A. Unterberger and J. Bokobza. Sur les opérateurs pseudo-différentiels d’ordre variable. C. R. Acad. Sci. Paris, 261:2271–2273, 1965.

    MathSciNet  MATH  Google Scholar 

  53. L. Waelbroeck. Le calcul symbolique dans les algébres commutatives. J. Math. Pures Appl., 33(9):147–186, 1954.

    MathSciNet  MATH  Google Scholar 

  54. H. Widom. A complete symbolic calculus for pseudodifferential operators. Bull. Sc. math.,2 e série, 104:19–63, 1980.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, Mainz, D-55099, Germany

    F. Baldus

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  1. F. Baldus
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Editors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Bonn, 53115, Germany

    Sergio Albeverio

  2. Institut für Mathematik, Technische Universität Clausthal, Clausthal-Zellerfeld, 38678, Germany

    Michael Demuth

  3. Institut für Mathematik, Universität Potsdam, Potsdam, 14415, Germany

    Elmar Schrohe  & Bert-Wolfgang Schulze  & 

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Baldus, F. (2003). An Approach to a Version of the S(M, g)-pseudo-differential Calculus on Manifolds. In: Albeverio, S., Demuth, M., Schrohe, E., Schulze, BW. (eds) Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations. Operator Theory: Advances and Applications, vol 145. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8073-2_4

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  • DOI: https://doi.org/10.1007/978-3-0348-8073-2_4

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9429-6

  • Online ISBN: 978-3-0348-8073-2

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