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

366 Pages·2002·6.872 MB·English

by  Thomas Krainer (auth.), Sergio Albeverio, Michael Demuth, Elmar Schrohe, Bert-Wolfgang Schulze (eds.)

Operator Theory: Advances and
Applications
Vol. 138
Editor:
I. Gohberg
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Subseries
Advances in Partial Differential
Equations
Subseries editors:
Bert-Wolfgang Schulze Sergio Albeverio
Institut fUr Mathematik Institut fOr Angewandte Mathematik
Universitat Potsdam Universitat Bonn
14415 Potsdam 53115 Bonn
Germany Germany
Michael Demuth Elmar Schrohe
Institut fUr Mathematik Institut fUr Mathematik
Technische Universitat Clausthal Universitat Potsdam
38678 Clausthal-Zellerfeld 14415 Potsdam
Germany Germany
Parabolicity,
Volterra Calculus, and
Conical Singularities
A Volume of
Advances in Partial Differential Equations
Sergio Albeverio
Michael Demuth
Elmar Schrohe
Bert-Wolfgang Schulze
Editors
Springer Basel AG
Editors:
Sergio Albeverio Elmar Schrohe
Institut fUr Angewandte Mathematik Institut fUr Mathematik
Universităt Bonn Universităt Potsdam
53115 Bonn 14415 Potsdam
Germany Germany
e-mail: [email protected] e-mail: [email protected]
Michael Demuth Bert-Wolfgang Schulze
Institut fUr Mathematik Institut fUr Mathematik
Technische Universităt Clausthal Universităt Potsdam
38678 Clausthal-Zellerfeld 14415 Potsdam
Germany Germany
e-mail: [email protected] e-mail: [email protected]
2000 Mathematics Subject Classification 47G30, 58J40
A CIP catalogue record for this book is available from the
Library ofCongress, Washington D.C., USA
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ISBN 978-3-0348-9469-2 ISBN 978-3-0348-8191-3 (eBook)
DOI 10.1007/978-3-0348-8191-3
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ISBN 978-3-0348-9469-2
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Contents
Preface.................................................................... xi
Volterra Families of Pseudodifferential Operators
Thomas K miner
Introduction ............................................................... 1
1. Basic notation and general conventions ................................ 4
1.1. Sets of real and complex numbers................................ 4
1.2. Multi-index notation ............................................. 5
1.3. Functional analysis and basic function spaces .................... 5
1.4. Tempered distributions and the Fourier transform ............... 6
2. General parameter-dependent symbols................................. 6
2.1. Asymptotic expansion........................................... 8
2.2. Homogeneity and classical symbols............................... 11
3. Parameter-dependent Volterra symbols ................................ 14
3.1. Kernel cut-off and asymptotic expansion......................... 14
3.2. The translation operator in Volterra symbols ..................... 25
4. The calculus of pseudodifferential operators.. . . .. . . . . .. . . .. . .. . ... .. .. . 28
4.1. Elements of the calculus......................................... 29
4.2. The formal adjoint operator ..................................... 33
4.3. Sobolev spaces and continuity ................................... 34
4.4. Coordinate invariance ........................................... 36
5. Ellipticity and parabolicity ............................................ 39
5.1. Ellipticity in the general calculus ................................ 39
5.2. Parabolicity in the Volterra calculus ............................. 41
References ................................................................. 43
The Calculus of Volterra Mellin Pseudodifferential Operators with
Operator-valued Symbols
Thomas K miner
Introduction. . . . .. . . .. .. . . . . . . . .. .... ... . .. . . . . . .. . . .. ... . .. ... .. .. ... . . .. . 47
1. Preliminaries on function spaces and the Mellin transform ............. 50
1.1. A Paley-Wiener type theorem.................................... 51
1.2. The Mellin transform in distributions ............................ 52
2. The calculus of Volterra symbols ....................................... 53
VI Contents
2.1. General anisotropic and Volterra symbols ........................ 53
2.1.1. Hilbert spaces with group-actions. ....................... 53
2.1.2. Definition of the symbol spaces. .......................... 53
2.1.3. Asymptotic expansion. ................................... 55
2.1.4. The translation operator in Volterra symbols. ............ 55
2.2. Holomorphic Volterra symbols ................................... 57
3. The calculus of Volterra Mellin operators .............................. 59
3.1. General Volterra Mellin operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2. Continuity in Mellin Sobolev spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3. Volterra Mellin operators with analytic symbols .................. 69
4. Kernel cut-off and Mellin quantization ................................. 74
4.1. The Mellin kernel cut-off operator ............................... 74
4.2. Degenerate symbols and Mellin quantization..................... 76
5. Parabolicity and Volterra parametrices ................................ 82
5.1. Ellipticity and parabolicity on symbolic level. . . . . . . . . . . . . . . . . . . .. 82
5.2. The parametrix construction..................................... 86
References ................................................................. 89
On the Inverse of Parabolic Systems of Partial Differential Equations
of General Form in an Infinite Space-Time Cylinder
Thomas Krainer and Bert-Wolfgang Schulze
Introduction ............................................................... 93
Chapter 1. Preliminary material ......................................... 103
1.1. Basic notation and general conventions .............................. 103
Functional analysis and basic function spaces ........................ 105
Preliminaries on function spaces and the Mellin transform ........... 107
Global analysis ...................................................... 109
1.2. Finitely meromorphic Fredholm families in \II-algebras ............... 110
1.3. Volterra integral operators ........................................... 118
Some notes on abstract kernels ...................................... 121
Chapter 2. Abstract Volterra pseudo differential calculus ................. 123
2.1. Anisotropic parameter-dependent symbols ........................... 123
Asymptotic expansion ............................................... 125
Classical symbols .................................................... 126
2.2. Anisotropic parameter-dependent operators .......................... 127
Elements of the calculus ............................................. 128
Ellipticity and parametrices .......................................... 130
Sobolev spaces and continuity ....................................... 132
Coordinate invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 133
Contents vii
2.3. Parameter-dependent Volterra symbols ............................... 134
Kernel cut-off and asymptotic expansion of Volterra symbols ......... 135
The translation operator in Volterra symbols ......................... 137
2.4. Parameter-dependent Volterra operators ............................. 138
Elements of the calculus ............................................. 139
Continuity and coordinate invariance ................................ 140
Parabolicity for Volterra pseudodifferential operators ................. 141
2.5. Volterra Mellin calculus .............................................. 144
Continuity in Mellin Sobolev spaces .................................. 148
2.6. Analytic Volterra Mellin calculus .................................... 149
Elements of the calculus ............................................. 152
The Mellin kernel cut-off operator and asymptotic expansion ......... 153
Degenerate symbols and Mellin quantization ......................... 155
2.7. Volterra Fourier operators with global weight conditions ............. 158
Chapter 3. Parameter-dependent Volterra calculus
on a closed manifold ....................................................... 161
3.1. Anisotropic parameter-dependent operators .......................... 161
Ellipticity and parametrices .......................................... 168
3.2. Parameter-dependent Volterra operators ............................. 170
Kernel cut-off behaviour and asymptotic expansion .................. 173
The translation operator in Volterra pseudodifferential operators ..... 175
Parabolicity for Volterra operators on manifolds ..................... 176
Chapter 4. Weighted Sobolev spaces ..................................... 178
4.1. Anisotropic Sobolev spaces on the infinite cylinder ................... 178
4.2. Anisotropic Mellin Sobolev spaces ................................... 181
Mellin Sobolev spaces with asymptotics .............................. 184
4.3. Cone Sobolev spaces ................................................. 187
Chapter 5. Calculi built upon parameter-dependent operators ............ 191
5.1. Anisotropic meromorphic Mellin symbols ............................ 191
5.2. Meromorphic Volterra Mellin symbols ................................ 198
Mellin quantization .................................................. 201
5.3. Elements of the Mellin calculus ...................................... 202
Ellipticity and Parabolicity .......................................... 205
5.4. Elements of the Fourier calculus with global weights ................. 211
Ellipticity and Parabolicity .......................................... 214
Chapter 6. Volterra cone calculus ........................................ 219
6.1. Green operators ..................................................... 219
6.2. The algebra of conormal operators ................................... 224
Operators that generate asymptotics ................................. 224
Calculus of conormal symbols ........................................ 225
The operator calculus ................................................ 228
Smoothing Mellin and Green operators .............................. 240
viii Contents
6.3. The algebra of Volterra cone operators ............................... 242
The symbolic structure .............................................. 251
Compositions and adjoints ........................................... 253
6.4. Ellipticity and Parabolicity .......................................... 259
Parabolic reductions of orders ....................................... 268
Chapter 7. Remarks on the classical theory
of parabolic PDE .......................................................... 270
References ................................................................. 275
On the Factorization of Meromorphic Mellin Symbols
Ingo Witt
1. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 279
2. Preliminaries .......................................................... 281
2.1. Parameter-dependent operators .................................. 281
2.2. Meromorphic Mellin symbols .................................... 285
2.3. Reduction to holomorphic Mellin symbols ........................ 286
3. Logarithms of pseudodifferential operators ............................. 287
3.1. The classes £IdOg(X; A) .......................................... 287
3.2. The exponential map ............................................ 291
3.3. The topological invariant \II(A) .................................. 296
3.4. Characterization of the image of exp ............................. 298
4. The kernel cut-off technique ........................................... 300
5. Proof of the main theorem ............................................. 302
5.1. Beginning of the proof ........................................... 302
5.2. Continuation of the proof ........................................ 303
5.3. The remaining case for dim X = 1 ............................... 303
References ................................................................. 305
Coordinate Invariance of the Cone Algebra with Asymptotics
David Kapanadze, Bert-Wolfgang Schulze, and Ingo Witt
Introduction ............................................................... 307
1. Cone operators on the half-axis ........................................ 309
1.1. The cone algebra ................................................ 309
1.2. Spaces with asymptotics and Green operators .................... 313
1.3. Push-forward of Mellin operators ................................ 315
1.4. Invariance ofthe cone algebra ................................... 321
Appendix to Sect. 1.4: An intrinsic interpretation
of the principal symbol .......................................... 322
1.5. Symbolic rules ................................................... 326
Contents ix
2. Operators on higher-dimensional cones ................................. 342
2.1. The cone algebra ................................................ 342
2.2. Spaces with asymptotics and Green operators .................... 346
2.3. Push-forward of Mellin operators ................................ 348
2.4. Invariance of the cone algebra ................................... 353
References ................................................................. 357
Preface
Partial differential equations constitute an integral part of mathematics. They lie
at the interface of areas as diverse as differential geometry, functional analysis, or
the theory of Lie groups and have numerous applications in the applied sciences.
A wealth of methods has been devised for their analysis. Over the past
decades, operator algebras in connection with ideas and structures from geometry,
topology, and theoretical physics have contributed a large variety of particularly
useful tools.
One typical example is the analysis on singular configurations, where elliptic
equations have been studied successfully within the framework of operator algebras
with symbolic structures adapted to the geometry of the underlying space. More
recently, these techniques have proven to be useful also for studying parabolic
and hyperbolic equations. Moreover, it turned out that many seemingly smooth,
noncompact situations can be handled with the ideas from singular analysis.
The three papers at the beginning of this volume highlight this aspect. They
deal with parabolic equations, a topic relevant for many applications. The first
article prepares the ground by presenting a calculus for pseudo differential operators
with an anisotropic analytic parameter. In the subsequent paper, an algebra of
Mellin operators on the infinite space-time cylinder is constructed. 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 Mellin 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.
This book is the sixth title of "Advances in Partial Differential Equations" , a
series originating from the work of the research group "Partial Differential Equa
tions and Complex Analysis" at the University of Potsdam. It is our intention to
promote expositions of a systematic character, written by specialists who report
on recent progress and give specific insight into their own work.
Bonn, Clausthal, and Potsdam, March 2002
S. Albeverio M. Demuth E. Schrohe B.-W. Schulze