Advertisement

Annals of Global Analysis and Geometry

, Volume 14, Issue 4, pp 403–425 | Cite as

Invariance of the cone algebra without asymptotics

  • Elmar Schrohe
Article

Abstract

Let B be a manifold with conical singularities, and denote by ℬ the smooth bounded manifold with cylindrical ends obtained by blowing up near the singularities.

B.-W. Schulze has developed a framework for a pseudodifferential calculus on B by defining various classes of distribution spaces and operator algebras, working in fixed coordinates on the manifold ℬ. I am showing here that the Mellin Sobolev spaces without asymptotics, the cone algebra without asymptotics, and its ideal of smoothing operators are independent of the choice of coordinates and therefore may be considered intrinsic objects for manifolds with conical singularities.

Key words

Manifolds with conical singularities Mellin calculus pseudodifferential operators 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Agranovič, M.S.; Višik, M.I.: Elliptic problems with a parameter and parabolic problems of general type (Russ.). Usp. Mat. Nauk 19 (1963), 53–161 (Engl. transl. Russ. Math. Surveys 19 (1963), 53–159).Google Scholar
  2. [2]
    Dieudonné, J.: Grundzüge der modernen Analysis, vol. 7. VEB Deutscher Verlag der Wissenchaften, Berlin 1982.Google Scholar
  3. [3]
    Egorov, Yu.; Schulze, B.-W.: Pseudo-Differential Operators, Singularities, Applications. Birkhäuser, Boston, Basel, Berlin (to appear).Google Scholar
  4. [4]
    Eichhorn, J.: Elliptic differential operators on noncompact manifolds. In: Schulze, B.-W.; Triebel, H. (Eds.): Seminar Analysis of the Karl-Weierstra\-Institute 1986/87. Teubner-Texte Math. 106, Leipzig 1988, 4–169.Google Scholar
  5. [5]
    Kondrat'ev, V.A.: Boundary value problems in domains with conical or angular points. Trans. Mosc. Math. Soc. 16 (1967), 227–313.Google Scholar
  6. [6]
    Lewis, J.E.; Parenti, C.: Pseudodifferential operators of Mellin type. Commun. Partial Differ. Equations 8 (1983), 477–544.Google Scholar
  7. [7]
    Melrose, R.: Transformation of boundary problems. Acta Math. 147 (1981), 149–236.Google Scholar
  8. [8]
    Melrose, R.: The Atiyah-Patodi-Singer Index Theorem. A K Peters, Wellesley, MA, 1993.Google Scholar
  9. [9]
    Plamenevskij, B. A.:Algebras of Pseudodifferential Operators (Russ.). Nauka, Moscow 1986.Google Scholar
  10. [10]
    Rempel, S.; Schulze, B.-W.: Complete Mellin and Green symbolic calculus in spaces with conormal asymptotics. Ann. Global Anal. Geom. 4 (1986), 137–224.Google Scholar
  11. [11]
    Schrohe, E.: Coordinate Invariance of the Mellin Calculus without Asymptotics for Manifolds with Conical Singularities. To appear in: Structure of solutions for partial differential equations, Kyoto 1994. RIMS Koukyuuroku (Kyoto University).Google Scholar
  12. [12]
    Schrohe, E.; Schulze, B.-W.: Boundary value problems in Boutet de Monvel's singularities I. In: Demuth, M.; Schrohe, E.; Schulze, B.-W. (Eds.): Pseudodifferential Operators and Mathematical Physics. Adv. Partial Differ. Equations 1. Akademie Verlag, Berlin 1994, 97–209.Google Scholar
  13. [13]
    Schrohe, E.; Schulze, B.-W.: Boudary value problems in Boutet de Monvel's algebra for manifolds with conical singularities II. In: Demuth, M.; Schrohe, E.; Schulze, B.-W. (Eds.): Boundary Value Problems, Deformation Quantization, Schrödinger Operators. Adv. Partial Differ. Equations 2. Akademie Verlag, Berlin 1995, 70–205.Google Scholar
  14. [14]
    Schulze, B.-W.: Pseudo-Differential Operators on Manifolds with Singularities. North-Holland, Amsterdam 1991.Google Scholar
  15. [15]
    Schulze, B.-W.: Mellin representation of pseudo-differential operators on manifolds with corners. Ann. Global Anal. Geom. 8 (1990), 261–297.Google Scholar
  16. [16]
    Schulze, B.-W.: Pseudo-Differential Boundary Value Problems, Conical Singularities and Asymptotics. Akademie Verlag, Berlin 1994.Google Scholar
  17. [17]
    Schulze, B.-W.: Transmission algebras on singular spaces with components of different dimensions. In: Demuth, M.; Schulze, B.-W. (Eds.): Partial Differential Operators and Mathematical Physics. Birkhäuser, Boston, Basel, Berlin 1995.Google Scholar
  18. [18]
    Višik, M.I.; Eskin, G.I.: Convolution equations in bounded domains in spaces with weighted norms. Math. USSR, Sb. 69 (1966), 65–110.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Elmar Schrohe
    • 1
  1. 1.Max-Planck-Arbeitsgruppe “Partielle Differentialgleichungen and Komplexe Analysis”Universität PotsdamPotsdamGermany

Personalised recommendations