On the asymptotic completeness of the Volterra calculus
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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.
KeywordsHeat Kernel Pseudodifferential Operator Conical Singularity Smoothing Operator Spectral Triple
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- [BG] R. Beals and P. Greiner,Calculus on Heisenberg manifolds, Ann. of Math. Stud.119, Princeton Univ. Press, 1988.Google Scholar
- [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
- [Kr1] T. Krainer,Parabolic pseudodifferetial operators and long-time asymptotics of solutions, PhD dissertation, University of Potsdam, 2000.Google Scholar
- [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
- [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
- [MSV] R. B. Melrose, I. M. Singer and M. Varghese,Fractional analytic index, E-print, arXiv, February 2004.Google Scholar
- [Mi2] H. Mikayelyan,Parabolic boundary and transmission problems, PhD dissertation, University of Potsdam, 2003.Google Scholar
- [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
- [Po3] R. Ponge,Hypoelliptic functional calculus on Heisenberg calculus. I, E-print, arXiv, September 2004.Google Scholar
- [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
- [Ta] M. E. Taylor,Noncommutative Microlocal Analysis. I, Mem. Amer. Math. Soc.52 (1984), no. 313.Google Scholar