Operator Theory: Advances and Applications Vol. 138 Editor: I. Gohberg Editorial Office: School of Mathematical H.G. Kaper (Argonne) Sciences S.T. Kuroda (Tokyo) Tel Aviv University P. Lancaster (Calgary) Ramat Aviv, Israel L.E. Lerer (Haifa) B. Mityagin (Columbus) Editorial Board: V. V. Peller (Manhattan, Kansas) J. Arazy (Haifa) M. Rosenblum (Charlottesville) A. Atzmon (Tel Aviv) J. Rovnyak (Charlottesville) J. A. Ball (Blacksburg) D. E. Sarason (Berkeley) A. Ben-Artzi (Tel Aviv) H. Upmeier (Marburg) H. Bercovici (Bloomington) S. M. Verduyn Lunel (Leiden) A. Bottcher (Chemnitz) D. Voiculescu (Berkeley) K. Clancey (Athens, USA) H. Widom (Santa Cruz) L. A. Coburn (Buffalo) D. Xia (Nashville) K. R. Davidson (Waterloo, Ontario) D. Yafaev (Rennes) R. G. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) P. A. Fillmore (Halifax) Honorary and Advisory P. A. Fuhrmann (Beer Sheva) Editorial Board: S. Goldberg (College Park) C. Foias (Bloomington) B. Gramsch (Mainz) P. R. Halmos (Santa Clara) G. Heinig (Chemnitz) T. Kailath (Stanford) J. A. Helton (La Jolla) P. D. Lax (New York) M.A. Kaashoek (Amsterdam) M. S. Livsic (Beer Sheva) 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 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 978-3-0348-9469-2 ISBN 978-3-0348-8191-3 (eBook) DOI 10.1007/978-3-0348-8191-3 This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of iIIustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind ofuse permission ofthe copyright owner must be obtained. © 2002 Springer Basel AG Originally published by Birkhauser Verlag, Basel -Boston -Berlin in 2002 Printed on acid-free paper produced from chlorine-free pulp. TCF 00 Cover design: Heinz Hiltbrunner, Basel ISBN 978-3-0348-9469-2 987654321 www.birkhasuer-science.com 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