Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the asymptotic completeness of the Volterra calculus

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.

This is a preview of subscription content, log in to check access.

References

  1. [AG] S. Alinhac and P. Gérard,Opérateurs pseudo-différentiels et théorème de Nash-Moser, InterEditions, Paris and Editions du CNRS, Meudon, 1991.

  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.

  3. [AS] M. Atiyah and I. Singer,The index of elliptic operators III, Ann. of Math. (2)87 (1968), 546–604.

  4. [BG] R. Beals and P. Greiner,Calculus on Heisenberg manifolds, Ann. of Math. Stud.119, Princeton Univ. Press, 1988.

  5. [BGS] R. Beals, P. Greiner and N. Stanton,The heat equation on a CR manifold, J. Differential Geom.20 (1984), 343–387.

  6. [BS] T. Buchholz and B. W. Schulze,Anisotropic edge pseudo-differential operators with discrete asymptotics, Math. Nachr.184 (1997), 73–125.

  7. [CM] A. Connes and H. Moscovici,The local index formula in noncommutative geometry, Geom. Funct. Anal.5 (1995), 174–243.

  8. [Ge] E. Getzler,A short proof of the local Atiyah-Singer index theorem, Topology25 (1986), 111–117.

  9. [Gr] P. Greiner,An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal.41 (1971), 163–218.

  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.

  11. [Kr1] T. Krainer,Parabolic pseudodifferetial operators and long-time asymptotics of solutions, PhD dissertation, University of Potsdam, 2000.

  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.

  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.

  14. [Me] R. Melrose,The Atiyah-Patodi-Singer Index Theorem, A.K. Peters, Wellesley, MA, 1993.

  15. [MSV] R. B. Melrose, I. M. Singer and M. Varghese,Fractional analytic index, E-print, arXiv, February 2004.

  16. [Mi1] H. Mikayelyan,Asymptotic summation of operator-valued Volterra symbols, J. Contemp. Math. Anal.37 (2002), 76–80.

  17. [Mi2] H. Mikayelyan,Parabolic boundary and transmission problems, PhD dissertation, University of Potsdam, 2003.

  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.

  19. [Pi1] A. Piriou,Une classe d'opérateurs pseudo-différentiels du type de Volterra, Ann. Inst. Fourier20 (1970), 77–94.

  20. [Pi2] A. Piriou,Problèmes aux limites généraux pour des opérateurs différentiels paraboliques dans un domaine borné, Ann. Inst. Fourier21 (1971), 59–78.

  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.

  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.

  23. [Po3] R. Ponge,Hypoelliptic functional calculus on Heisenberg calculus. I, E-print, arXiv, September 2004.

  24. [Sh] M. Shubin,Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987.

  25. [Sc1] B. W. Schulze,Pseudo-Differential Operators on Manifolds with Singularities, North-Holland, Amsterdam, 1991.

  26. [Sc2] B. W. Schulze,Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, Chichester, 1998.

  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.

  28. [Ta] M. E. Taylor,Noncommutative Microlocal Analysis. I, Mem. Amer. Math. Soc.52 (1984), no. 313.

Download references

Author information

Correspondence to Raphaël Ponge.

Additional information

The author was partially supported by the European RT NetworkGeometric Analysis HPCRN-CT-1999-00118.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ponge, R., Mikayelyan, H. On the asymptotic completeness of the Volterra calculus. J. Anal. Math. 94, 249–263 (2004). https://doi.org/10.1007/BF02789049

Download citation

Keywords

  • Heat Kernel
  • Pseudodifferential Operator
  • Conical Singularity
  • Smoothing Operator
  • Spectral Triple