Parabolicity, Volterra Calculus, and Conical Singularities : A Volume of Advances in Partial Differential Equations

by Albeverio, Sergio; Demuth, Michael; Schrohe, Elmar; Schulze, Bert-Wolfgang

• ISBN13: 9783764369064
• ISBN10: 376436906X
• Format: Hardcover
• Copyright: 2003-03-01
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Preface	xi
Volterra Families of Pseudodifferential Operators
Thomas Krainer
Introduction	1	(3)
Basic notation and general conventions	4	(2)
Sets of real and complex numbers	4	(1)
Multi-index notation	5	(1)
Functional analysis and basic function spaces	5	(1)
Tempered distributions and the Fourier transform	6	(1)
General parameter-dependent symbols	6	(8)
Asymptotic expansion	8	(3)
Homogeneity and classical symbols	11	(3)
Parameter-dependent Volterra symbols	14	(14)
Kernel cut-off and asymptotic expansion	14	(11)
The translation operator in Volterra symbols	25	(3)
The calculus of pseudodifferential operators	28	(11)
Elements of the calculus	29	(4)
The formal adjoint operator	33	(1)
Sobolev spaces and continuity	34	(2)
Coordinate invariance	36	(3)
Ellipticity and parabolicity	39	(11)
Ellipticity in the general calculus	39	(2)
Parabolicity in the Volterra calculus	41	(2)
References	43	(4)
The Calculus of Volterra Mellin Pseudodifferential Operators with Operator-valued Symbols
Thomas Krainer
Introduction	47	(3)
Preliminaries on function spaces and the Mellin transform	50	(3)
A Paley-Wiener type theorem	51	(1)
The Mellin transform in distributions	52	(1)
The calculus of Volterra symbols	53	(6)
General anisotropic and Volterra symbols	53	(1)
Hilbert spaces with group-actions	53	(1)
Definition of the symbol spaces	53	(2)
Asymptotic expansion	55	(1)
The translation operator in Volterra symbols	55	(2)
Holomorphic Volterra symbols	57	(2)
The calculus of Volterra Mellin operators	59	(15)
General Volterra Mellin operators	59	(8)
Continuity in Mellin Sobolev spaces	67	(2)
Volterra Mellin operators with analytic symbols	69	(5)
Kernel cut-off and Mellin quantization	74	(8)
The Mellin kernel cut-off operator	74	(2)
Degenerate symbols and Mellin quantization	76	(6)
Parabolicity and Volterra parametrices	82	(11)
Ellipticity and parabolicity on symbolic level	82	(4)
The parametrix construction	86	(3)
References	89	(4)
On the Inverse of Parabolic Systems of Partial Differential Equations of General Form in an Infinite Space-Time Cylinder
Thomas Krainer
Bert-Wolfgang Schulze
Introduction	93	(10)
Preliminary material	103	(20)
Basic notation and general conventions	103	(7)
Functional analysis and basic function spaces	105	(2)
Preliminaries on function spaces and the Mellin transform	107	(2)
Global analysis	109	(1)
Finitely meromorphic Fredholm families in Ψ-algebras	110	(8)
Volterra integral operators	118	(5)
Some notes on abstract kernels	121	(2)
Abstract Volterra pseudodifferential calculus	123	(38)
Anisotropic parameter-dependent symbols	123	(4)
Asymptotic expansion	125	(1)
Classical symbols	126	(1)
Anisotropic parameter-dependent operators	127	(7)
Elements of the calculus	128	(2)
Ellipticity and parametrices	130	(2)
Sobolev spaces and continuity	132	(1)
Coordinate invariance	133	(1)
Parameter-dependent Volterra symbols	134	(4)
Kernel cut-off and asymptotic expansion of Volterra symbols	135	(2)
The translation operator in Volterra symbols	137	(1)
Parameter-dependent Volterra operators	138	(6)
Elements of the calculus	139	(1)
Continuity and coordinate invariance	140	(1)
Parabolicity for Volterra pseudodifferential operators	141	(3)
Volterra Mellin calculus	144	(5)
Continuity in Mellin Sobolev spaces	148	(1)
Analytic Volterra Mellin calculus	149	(9)
Elements of the calculus	152	(1)
The Mellin kernel cut-off operator and asymptotic expansion	153	(2)
Degenerate symbols and Mellin quantization	155	(3)
Volterra Fourier operators with global weight conditions	158	(3)
Parameter-dependent Volterra calculus on a closed manifold	161	(17)
Anisotropic parameter-dependent operators	161	(9)
Ellipticity and parametrices	168	(2)
Parameter-dependent Volterra operators	170	(8)
Kernel cut-off behaviour and asymptotic expansion	173	(2)
The translation operator in Volterra pseudodifferential operators	175	(1)
Parabolicity for Volterra operators on manifolds	176	(2)
Weighted Sobolev spaces	178	(13)
Anisotropic Sobolev spaces on the infinite cylinder	178	(3)
Anisotropic Mellin Sobolev spaces	181	(6)
Mellin Sobolev spaces with asymptotics	184	(3)
Cone Sobolev spaces	187	(4)
Calculi built upon parameter-dependent operators	191	(28)
Anisotropic meromorphic Mellin symbols	191	(7)
Meromorphic Volterra Mellin symbols	198	(4)
Mellin quantization	201	(1)
Elements of the Mellin calculus	202	(9)
Ellipticity and Parabolicity	205	(6)
Elements of the Fourier calculus with global weights	211	(8)
Ellipticity and Parabolicity	214	(5)
Volterra cone calculus	219	(51)
Green operators	219	(5)
The algebra of conormal operators	224	(18)
Operators that generate asymptotics	224	(1)
Calculus of conormal symbols	225	(3)
The operator calculus	228	(12)
Smoothing Mellin and Green operators	240	(2)
The algebra of Volterra cone operators	242	(17)
The symbolic structure	251	(2)
Compositions and adjoints	253	(6)
Ellipticity and Parabolicity	259	(11)
Parabolic reductions of orders	268	(2)
Remarks on the classical theory of parabolic PDE	270	(87)
References	275	(4)
On the Factorization of Meromorphic Mellin Symbols
Ingo Witt
Introduction	279	(2)
Preliminaries	281	(6)
Parameter-dependent operators	281	(4)
Meromorphic Mellin symbols	285	(1)
Reduction to holomorphic Mellin symbols	286	(1)
Logarithms of pseudodifferential operators	287	(13)
The classes Lμlog(X; Λ)	287	(4)
The exponential map	291	(5)
The topological invariant Ψ(A)	296	(2)
Characterization of the image of exp	298	(2)
The kernel cut-off technique	300	(2)
Proof of the main theorem	302	(7)
Beginning of the proof	302	(1)
Continuation of the proof	303	(1)
The remaining case for dim X = 1	303	(2)
References	305	(2)
Coordinate Invariance of the Cone Algebra with Asymptotics
David Kapanadze
Bert-Wolfgang Schulze
Ingo Witt
Introduction	307	(2)
Cone operators on the half-axis	309	(33)
The cone algebra	309	(4)
Spaces with asymptotics and Green operators	313	(2)
Push-forward of Mellin operators	315	(6)
Invariance of the cone algebra	321	(1)
An intrinsic interpretation of the principal symbol	322	(4)
Symbolic rules	326	(16)
Operators on higher-dimensional cones	342	(15)
The cone algebra	342	(4)
Spaces with asymptotics and Green operators	346	(2)
Push-forward of Mellin operators	348	(5)
Invariance of the cone algebra	353	(4)
References	357