Advertisement

Journal d’Analyse Mathematique

, Volume 94, Issue 1, pp 249–263 | Cite as

On the asymptotic completeness of the Volterra calculus

  • Raphaël Ponge
  • H. Mikayelyan
Article

Abstract

The Volterra calculus is a simple and powerful pseudodifferential tool for inverting parabolic equations which has found many applications in geometric analysis. An important property in the theory of pseudodifferential operators is asymptotic completeness, which allows the construction of parametrices modulo smoothing operators. In this paper, we present new and fairly elementary proofs of the asymptotic completeness of the Volterra calculus.

Keywords

Heat Kernel Pseudodifferential Operator Conical Singularity Smoothing Operator Spectral Triple 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AG] S. Alinhac and P. Gérard,Opérateurs pseudo-différentiels et théorème de Nash-Moser, InterEditions, Paris and Editions du CNRS, Meudon, 1991.MATHGoogle Scholar
  2. [APS] M. F. Atiyah, V. K. Patodi and I. M. Singer,Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Philos. Soc.77 (1975), 43–69.MATHMathSciNetCrossRefGoogle Scholar
  3. [AS] M. Atiyah and I. Singer,The index of elliptic operators III, Ann. of Math. (2)87 (1968), 546–604.CrossRefMathSciNetGoogle Scholar
  4. [BG] R. Beals and P. Greiner,Calculus on Heisenberg manifolds, Ann. of Math. Stud.119, Princeton Univ. Press, 1988.Google Scholar
  5. [BGS] R. Beals, P. Greiner and N. Stanton,The heat equation on a CR manifold, J. Differential Geom.20 (1984), 343–387.MATHMathSciNetGoogle Scholar
  6. [BS] T. Buchholz and B. W. Schulze,Anisotropic edge pseudo-differential operators with discrete asymptotics, Math. Nachr.184 (1997), 73–125.MATHCrossRefMathSciNetGoogle Scholar
  7. [CM] A. Connes and H. Moscovici,The local index formula in noncommutative geometry, Geom. Funct. Anal.5 (1995), 174–243.MATHCrossRefMathSciNetGoogle Scholar
  8. [Ge] E. Getzler,A short proof of the local Atiyah-Singer index theorem, Topology25 (1986), 111–117.MATHCrossRefMathSciNetGoogle Scholar
  9. [Gr] P. Greiner,An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal.41 (1971), 163–218.MATHCrossRefMathSciNetGoogle Scholar
  10. [Hö] L. Hörmander,Pseudo-differential operators and hypoelliptic equations, inSingular Integrals (Proc. Sympos. Pure Math., Vol. X, 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 138–183.Google Scholar
  11. [Kr1] T. Krainer,Parabolic pseudodifferetial operators and long-time asymptotics of solutions, PhD dissertation, University of Potsdam, 2000.Google Scholar
  12. [Kr2] T. Krainer,Volterra families of pseudodifferential operators andThe calculus of Volterra Mellin pseudodifferential operators with operator-valued symbols, inParabolicity, Volterra Calculus, and Conical Singularities, Oper. Theory Adv. Appl.138, Birkhäuser, Basel, 2002, pp. 1–45 and 47–91.Google Scholar
  13. [KS] T. Krainer and B. W. Schulze,On the inverse of parabolic systems of partial differential equations of general form in an infinite space-time cylinder, inParabolicity, Volterra Calculus, and Conical Singularitiess, Birkhäuser, Basel, 2002, pp. 93–278.Google Scholar
  14. [Me] R. Melrose,The Atiyah-Patodi-Singer Index Theorem, A.K. Peters, Wellesley, MA, 1993.MATHGoogle Scholar
  15. [MSV] R. B. Melrose, I. M. Singer and M. Varghese,Fractional analytic index, E-print, arXiv, February 2004.Google Scholar
  16. [Mi1] H. Mikayelyan,Asymptotic summation of operator-valued Volterra symbols, J. Contemp. Math. Anal.37 (2002), 76–80.MathSciNetMATHGoogle Scholar
  17. [Mi2] H. Mikayelyan,Parabolic boundary and transmission problems, PhD dissertation, University of Potsdam, 2003.Google Scholar
  18. [Mit] M. Mitrea,The initial Dirichlet boundary value problem for general second order parabolic systems in nonsmooth manifolds, Comm. Partial Differential Equations26 (2001), 1975–2036.MATHCrossRefMathSciNetGoogle Scholar
  19. [Pi1] A. Piriou,Une classe d'opérateurs pseudo-différentiels du type de Volterra, Ann. Inst. Fourier20 (1970), 77–94.MATHMathSciNetGoogle Scholar
  20. [Pi2] A. Piriou,Problèmes aux limites généraux pour des opérateurs différentiels paraboliques dans un domaine borné, Ann. Inst. Fourier21 (1971), 59–78.MATHMathSciNetGoogle Scholar
  21. [Po1] R. Ponge,Calcul hypoelliptique sur les variétés de Heisenberg, résidu non commutatif et géométrie pseudo-hermitienne, PhD dissertation, University of Paris-Sud (Orsay), 2000.Google Scholar
  22. [Po2] R. Ponge,A new short proof of the local index formula and some of its applications, Comm. Math. Phys.241 (2003), 215–234.MATHCrossRefMathSciNetGoogle Scholar
  23. [Po3] R. Ponge,Hypoelliptic functional calculus on Heisenberg calculus. I, E-print, arXiv, September 2004.Google Scholar
  24. [Sh] M. Shubin,Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987.MATHGoogle Scholar
  25. [Sc1] B. W. Schulze,Pseudo-Differential Operators on Manifolds with Singularities, North-Holland, Amsterdam, 1991.MATHGoogle Scholar
  26. [Sc2] B. W. Schulze,Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, Chichester, 1998.MATHGoogle Scholar
  27. [Se] R. T. Seeley,Complex powers of an elliptic operator, inSingular Integrals (Proc. Sympos. Pure Math., Vol. X, 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307.Google Scholar
  28. [Ta] M. E. Taylor,Noncommutative Microlocal Analysis. I, Mem. Amer. Math. Soc.52 (1984), no. 313.Google Scholar

Copyright information

© The Hebrew University Magnes Press 2004

Authors and Affiliations

  1. 1.Department of MathematicsOhio State UniversityColumbusUSA
  2. 2.Mathematics InstituteUniversity of LeipzigLeipzigGermany

Personalised recommendations