Abstract
An informal overview is given of an algebra of pseudodifferential operators on manifolds with conical singularities as it was introduced by Schulze. It is proven that the residual class of Green operators, that by definition map Sobolev spaces to functions having certain prescribed asymptotics at the singularity, can equivalently be described as integral operators with smooth kernels, which have in both variables a corresponding asymptotic structure.
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M. Baranowski. Totally characteristic pseudo-differential operators in Besov-LizorkinTriebel spaces. Ann.Global Anal. Geom.7: 3–27, 1989.
B. Fedosov, B.-W. Schulze, N. Tarkhanov. A general index formula on toric manifolds with conical points. In J.B. Gil et al. (eds.)Approaches toSingular Analysis, Operator Theory: Advances and Applications, Birkhäuser, Basel, 2000.
J.B. Gil. Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators, preprint.
J.B. Gil, B.-W. Schulze, J. Seiler. Holomorphic operator-valued symbols for edge-degenerate pseudo-differential operators. In M. Demuth et al. (eds.)Differential Equations Asymptotic Analysis and Mathematical Physics Math. Research, volume 100, Akademie-Verlag, Berlin, 1997.
J.B. Gil, B.-W. Schulze, J. Seiler. Cone pseudodifferential operators in the edge symbolic calculus.Osaka J. Math.37: 221–260, 2000.
I.C. Gohberg, E.I. Sigal. An operator generalization of the logarithmic residue theorem and the theorem of Rouché.Math. USSR Sbornik13: 603–625, 1971.
Th. Krainer. Parabolic pseudo-differential operators and long-time asymptotics of solutions. Dissertation, Institut für Mathematik, Universität Potsdam. (submitted)
] P. Loya. The structure of the resolvent of elliptic pseudodifferential operators, preprint.
S. Rempel, B.-W. Schulze. Complete Mellin and Green symbolic calculus in spaces with conormal asymptotics.Ann. Global Anal. Geom.4: 137–224, 1986.
S. Rempel, B.-W. Schulze.Asymptotics for Elliptic MixedBoundaryProblems.Math. Research, volume 50 Akademie Verlag, Berlin, 1989.
E. Schrohe, B.-W. Schulze. Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities I. In M. Demuth et al. (eds.)Pseudo-differential Calculus and Mathematical PhysicsMath. Topics, volume 5: Advances in Partial Differential Equations, Akademie Verlag, Berlin, 1994.
E. Schrohe, B.-W. Schulze. Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities II. In M. Demuth et al. (eds.)Boundary Value Problems Schrödinger Operators Deformation QuantizationMath. Topics, volume 8: Advances in Partial Differential Equations, Akademie Verlag, Berlin, 1996.
E. Schrohe, J. Seiler. Ellipticity and invertibility in the cone algebra on LP-Sobolev spaces. Preprint 99/28, Institut für Mathematik, Universität Potsdam, 1999.
B.-W. Schulze. Mellin Expansions of pseudo-differential operators and conormal asymptotics of solutions. InProc. Oberwolfach1986, Lect. Notes in Math. 1256, Springer Verlag, Berlin, 1987.
B.-W. Schulze. The Mellin pseudo-differential calculus on manifolds with corners. InSymposium`Analysis onManifolds withSingularities’, Breitenbrunn1990, Teubner-Texte Math. 131, Teubner-Verlag, Stuttgart, 1992.
B.-W. Schulze.Pseudo-differential Operators on Manifolds with Singularities.North Holland, Amsterdam, 1991.
B.-W. Schulze.Pseudo-differentialBoundaryValue Problems,Conical Singularities,and Asymptotics.Math. Topics, volume 4, Akademie Verlag, Berlin, 1994.
B.-W. Schulze. BoundaryValue Problems and Singular Pseudo-Differential Operators.Wiley, Chichester, 1998.
J. Seiler. Pseudo-differential calculus on manifolds with non-compact edges. Dissertation, Institut für Mathematik, Universität Potsdam, 1997.
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Seiler, J. (2001). The Cone Algebra and a Kernel Characterization of Green Operators. In: Gil, J.B., Grieser, D., Lesch, M. (eds) Approaches to Singular Analysis. Operator Theory: Advances and Applications, vol 125. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8253-8_1
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DOI: https://doi.org/10.1007/978-3-0348-8253-8_1
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