## Parabolicity, Volterra Calculus, and Conical SingularitiesSergio Albeverio, Michael Demuth, Elmar Schrohe, Bert-Wolfgang Schulze 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 |

357 | |

### Other editions - View all

Parabolicity, Volterra Calculus, and Conical Singularities Sergio Albeverio,Michael Demuth,Elmar Schrohe No preview available - 2002 |

### Common terms and phrases

Analogously analytic anisotropic assertion asymptotic expansion asymptotic type belongs bilinear mappings Boundary Value Problems closed graph theorem compact complete symbol cone algebra conormal symbols Consequently continuous embedding coordinate cut-off functions defined denote diffeomorphism elliptic parabolic embedding exists following are equivalent following asymptotic expansion formal adjoint Fourier Frechet spaces Green operators half-plane Hilbert spaces holomorphic homogeneous principal symbol invertible kernel cut-off operator left-symbol Leibniz-product Lemma manifolds Math Mellin transform meromorphic Mellin symbols modulo Moreover neighbourhood Notation obtain operator-valued symbols oscillatory integral formula parameter parameter-dependent elliptic parametrix Partial Differential Equations particular Potsdam proof of Theorem properties Proposition pseudodifferential operators Remark respectively restricts Rn x H scales of Hilbert Schulze Section sequence smooth Sobolev spaces spaces with group-actions symbol spaces symbols of order Ti_7 vector bundle Verlag Volterra Mellin symbols Volterra operators Volterra symbols weight x Rn x