Parabolicity, Volterra Calculus, and Conical Singularities

Front Cover
Sergio Albeverio, Michael Demuth, Elmar Schrohe, Bert-Wolfgang Schulze
Springer Science & Business Media, 2002 - Mathematics - 358 pages
This volume highlights the analysis on noncompact and singular manifolds within the framework of the cone calculus with asymptotics. The three papers at the beginning deal with parabolic equations, a topic relevant for many applications. The first article presents a calculus for pseudodifferential operators with an anisotropic analytic parameter. The subsequent paper develops an algebra of Mellin operators on the infinite space-time cylinder. It is shown how timelike infinity can be treated as a conical singularity. In the third text - the central article of this volume - the authors use these results to obtain precise information on the long-time asymptotics of solutions to parabolic equations and to construct inverses within the calculus. There follows a factorization theorem for meromorphic symbols: It is proven that each of these can be decomposed into a holomorphic invertible part and a smoothing part containing all the meromorphic information. It is expected that this result will be important for applications in the analysis of nonlinear hyperbolic equations. The final article addresses the question of the coordinate invariance of the Mellin calculus with asymptotics.
 

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Contents

Parameterdependent Volterra symbols
14
The calculus of pseudodifferential operators
28
References
43
The calculus of Volterra Mellin operators
59
Kernel cutoff and Mellin quantization
74
Parabolicity and Volterra parametrices
82
References
89
Preliminary material
103
Calculi built upon parameterdependent operators
191
Volterra cone calculus
219
Remarks on the classical theory
270
On the Factorization of Meromorphic Mellin Symbols
279
Logarithms of pseudodifferential operators
287
The kernel cutoff technique
300
Introduction
307
Operators on higherdimensional cones
342

Abstract Volterra pseudodifferential calculus
123
Parameterdependent Volterra calculus
161
Weighted Sobolev spaces
178

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