Pseudo-Differential Operators Theory and Applications Vol. 2

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2 Pseudo-Differential Operators Theory and Applications Vol. 2 Managing Editor M.W. Wong (York University, Canada) Editorial Board Luigi Rodino (Università di Torino, Italy) Bert-Wolfgang Schulze (Universität Potsdam, Germany) Johannes Sjöstrand (École Polytechnique, Palaiseau, France) Sundaram Thangavelu (Indian Institute of Science at Bangalore, India) Marciej Zworski (University of California at Berkeley, USA) Pseudo-Differential Operators: Theory and Applications is a series of moderately priced graduate-level textbooks and monographs appealing to students and experts alike. Pseudo-differential operators are understood in a very broad sense and include such topics as harmonic analysis, PDE, geometry, mathematical physics, microlocal analysis, time-frequency analysis, imaging and computations. Modern trends and novel applications in mathematics, natural sciences, medicine, scientific computing, and engineering are highlighted.

3 Michael Ruzhansky Ville Turunen Pseudo-Differential Operators and Symmetries Background Analysis and Advanced Topics Birkhäuser Basel Boston Berlin

4 Authors: Michael Ruzhansky Department of Mathematics Imperial College London 180 Queen s Gate London SW7 2AZ United Kingdom m.ruzhansky@imperial.ac.uk Ville Turunen Institute of Mathematics Helsinki University of Technology P.O. Box 1100 FI TKK Finland ville.turunen@hut.fi 2000 Mathematics Subject Classification: 35Sxx, 58J40; 43A77, 43A80, 43A85 Library of Congress Control Number: 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 < ISBN Birkhäuser Verlag AG, Basel Boston Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained Birkhäuser Verlag AG Basel Boston Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF Printed in Germany ISBN e-isbn

5 Contents Preface... xiii Introduction... 1 Part I Foundations of Analysis A Sets, Topology and Metrics A.1 Sets,collections,families... 9 A.2 Relations,functions,equivalencesandorders A.3 Dominoestumblingandtransfiniteinduction A.4 AxiomofChoice:equivalentformulations A.5 Well-OrderingPrinciplerevisited A.6 Metricspaces A.7 Topologicalspaces A.8 Kuratowski sclosure A.9 Completemetricspaces A.10 Continuityandhomeomorphisms A.11 Compacttopologicalspaces A.12 CompactHausdorffspaces A.13 Sequentialcompactness A.14 Stone Weierstrasstheorem A.15 Manifolds A.16 Connectednessandpath-connectedness A.17 Co-inductionandquotientspaces A.18 Inductionandproductspaces A.19 Metrisabletopologies A.20 Topologyviageneralisedsequences... 77

6 vi Contents B C Elementary Functional Analysis B.1 Vectorspaces B.1.1 Tensorproducts B.2 Topologicalvectorspaces B.3 Locallyconvexspaces B.3.1 Topologicaltensorproducts B.4 Banachspaces B.4.1 Banachspaceadjoint B.5 Hilbertspaces B.5.1 Trace class, Hilbert Schmidt, and Schatten classes Measure Theory and Integration C.1 Measuresandoutermeasures C.1.1 Measuringsets C.1.2 Borelregularity C.1.3 OnLebesguemeasure C.1.4 Lebesguenon-measurablesets C.2 Measurablefunctions C.2.1 Well-behavingfunctions C.2.2 Sequencesofmeasurablefunctions C.2.3 Approximatingmeasurablefunctions C.3 Integration C.3.1 Integratingsimplenon-negativefunctions C.3.2 Integratingnon-negativefunctions C.3.3 Integrationingeneral C.4 Integralasafunctional C.4.1 Lebesgue spaces L p (μ) C.4.2 Signedmeasures C.4.3 Derivativesofsignedmeasures C.4.4 Integrationasfunctionalonfunctionspaces C.4.5 Integration as functional on L p (μ) C.4.6 Integration as functional on C(X) C.5 Productmeasureandintegral D Algebras D.1 Algebras D.2 Topologicalalgebras D.3 Banachalgebras D.4 CommutativeBanachalgebras D.5 C -algebras D.6 Appendix:Liouville stheorem

7 Contents vii Part II Commutative Symmetries 1 Fourier Analysis on R n 1.1 BasicpropertiesoftheFouriertransform Usefulinequalities Tempereddistributions Fouriertransformoftempereddistributions Operationswithdistributions Approximatingbysmoothfunctions Distributions Localisation of L p -spacesanddistributions Convolutionofdistributions Sobolevspaces WeakderivativesandSobolevspaces SomepropertiesofSobolevspaces Mollifiers ApproximationofSobolevspacefunctions Interpolation Pseudo-differential Operators on R n 2.1 Motivationanddefinition Amplitude representation of pseudo-differential operators Kernelrepresentationofpseudo-differentialoperators Boundedness on L 2 (R n ) Calculusofpseudo-differentialoperators Compositionformulae Changesofvariables Principalsymbolandclassicalsymbols Calculus proof of L 2 -boundedness Asymptoticsums Applicationstopartialdifferentialequations FreezingprincipleforPDEs Elliptic operators Sobolevspacesrevisited Periodic and Discrete Analysis 3.1 Distributions and Fourier transforms on T n and Z n Sobolev spaces H s (T n ) Discreteanalysistoolkit Calculusoffinitedifferences Discrete Taylor expansion and polynomials on Z n Severaldiscreteinequalities

8 viii Contents Linkingdifferencestoderivatives PeriodicTaylorexpansion Appendix:onoperatorsinBanachspaces Pseudo-differential Operators on T n 4.1 Toroidalsymbols Quantization of operators on T n Toroidalsymbols Toroidalamplitudes Pseudo-differentialoperatorsonSobolevspaces Kernelsofperiodicpseudo-differentialoperators Asymptoticsumsandamplitudeoperators Extensionoftoroidalsymbols Periodisationofpseudo-differentialoperators Symboliccalculus Operators on L 2 (T n )andsobolevspaces Elliptic pseudo-differential operators on T n Smoothnessproperties Anapplicationtoperiodicintegraloperators Toroidalwavefrontsets Fourierseriesoperators Boundedness of Fourier series operators on L 2 (T n ) Anapplicationtohyperbolicequations Commutator Characterisation of Pseudo-differential Operators 5.1 Euclideancommutatorcharacterisation Pseudo-differentialoperatorsonmanifolds Commutatorcharacterisationonclosedmanifolds Toroidalcommutatorcharacterisation Part III Representation Theory of Compact Groups 6 Groups 6.1 Introduction Groupswithouttopology Groupactionsandrepresentations Topological Groups 7.1 Topologicalgroups Representationsoftopologicalgroups Compactgroups

9 Contents ix 7.4 Haar measure and integral Integrationonquotientspaces Peter Weyldecompositionofrepresentations Fourierseriesandtrigonometricpolynomials Convolutions Characters Inducedrepresentations Linear Lie Groups 8.1 Exponentialmap NosmallsubgroupsforLie,please LiegroupsandLiealgebras Universalenvelopingalgebra CasimirelementandLaplaceoperator Hopf Algebras 9.1 Commutative C -algebras Hopfalgebras Part IV Non-commutative Symmetries 10 Pseudo-differential Operators on Compact Lie Groups 10.1 Introduction FourierseriesoncompactLiegroups Functionspacesontheunitarydual Spaces on the group G Spaces on the dual Ĝ p Spaces L (Ĝ) Symbolsofoperators Fullsymbols Conjugationpropertiesofsymbols Boundedness of operators on L 2 (G) TaylorexpansiononLiegroups Symboliccalculus Differenceoperators Commutatorcharacterisation Calculus Leibnizformula Boundedness on Sobolev spaces H s (G) SymbolclassesoncompactLiegroups Some properties of symbols of Ψ m (G)

10 x Contents Symbol classes Σ m (G) Fullsymbolsoncompactmanifolds Operator-valuedsymbols Example on the torus T n Appendix:integralkernels Fourier Analysis on SU(2) 11.1 Preliminaries:groupsU(1),SO(2),andSO(3) EuleranglesonSO(3) PartialderivativesonSO(3) InvariantintegrationonSO(3) GeneralpropertiesofSU(2) EulerangleparametrisationofSU(2) Quaternions QuaternionsandSU(2) QuaternionsandSO(3) InvariantintegrationonSU(2) Symplecticgroups LiealgebraanddifferentialoperatorsonSU(2) IrreducibleunitaryrepresentationsofSU(2) RepresentationsofSO(3) MatrixelementsofrepresentationsofSU(2) MultiplicationformulaeforrepresentationsofSU(2) LaplacianandderivativesofrepresentationsonSU(2) FourierseriesonSU(2)andonSO(3) Pseudo-differential Operators on SU(2) 12.1 SymbolsofoperatorsonSU(2) Symbols of +,, 0 and Laplacian L Differenceoperatorsforsymbols DifferenceoperatorsonSU(2) Differences for symbols of +,, 0 and Laplacian L Differences for aσ SymbolclassesonSU(2) Pseudo-differential operators on S Appendix:infinitematrices

11 Contents xi 13 Pseudo-differential Operators on Homogeneous Spaces 13.1 Analysisonclosedmanifolds Analysisoncompacthomogeneousspaces Analysis on K\G, K atorus Liftingofoperators Bibliography Notation Index

12 Preface This monograph is devoted to the development of the theory of pseudo-differential operators on spaces with symmetries. Such spaces are the Euclidean space R n,the torus T n, compact Lie groups and compact homogeneous spaces. The book consists of several parts. One of our aims has been not only to present new results on pseudo-differential operators but also to show parallels between different approaches to pseudo-differential operators on different spaces. Moreover, we tried to present the material in a self-contained way to make it accessible for readers approaching the material for the first time. However, different spaces on which we develop the theory of pseudo-differential operators require different backgrounds. Thus, while operators on the Euclidean space in Chapter 2 rely on the well-known Euclidean Fourier analysis, pseudo-differential operators on the torus and more general Lie groups in Chapters 4 and 10 require certain backgrounds in discrete analysis and in the representation theory of compact Lie groups, which we therefore present in Chapter 3 and in Part III, respectively. Moreover, anyone who wishes to work with pseudo-differential operators on Lie groups will certainly benefit from a good grasp of certain aspects of representation theory. That is why we present the main elements of this theory in Part III, thus eliminating the necessity for the reader to consult other sources for most of the time. Similarly, the backgrounds for the theory of pseudo-differential operators on S 3 and SU(2) developed in Chapter 12 can be found in Chapter 11 presented in a self-contained way suitable for immediate use. However, it was still not a simple matter to make a self-contained presentation of these theories without referring to basics of the more general analysis. Thus, in hoping that this monograph may serve as a guide to different aspects of pseudodifferential operators, we decided to include the basics of analysis that are certainly useful for anyone working with pseudo-differential operators. Overall, we tried to supplement all the material with exercises for learning the ideas and practicing the techniques. They range from elementary problems to more challenging ones. In fact, on many occasions where other authors could say it is easy to see or one can check, we prefer to present it as an exercise. At the same time, more challenging exercises also serve as an excellent way to present more aspects of the discussed material.

13 xiv Preface We would like to thank Professor G. Vainikko, who introduced V. Turunen to pseudo-differential equations on circles [137], leading naturally to the noncommutative setting of the doctoral thesis. The thesis work was crucially influenced by a visit to M.E. Taylor in spring We are grateful to Professor M.W. Wong for suggesting that we write this monograph, to our students for giving us useful feedback on the background material of the book, and to Dr. J. Wirth for reading the manuscript and for his useful feedback and numerous comments, which led to clarifications of the presentation, especially of the material from Section Most of the work was carried out at the pleasant atmospheres provided by Helsinki University of Technology and Imperial College London. Moreover, over the years, we have outlined substantial parts of the monograph elsewhere: particularly, we appreciate the hospitality of University of North Carolina at Chapel Hill, University of Torino and Osaka University. The work of M. Ruzhansky was supported in part by EPSRC grants EP/E062873/01 and EP/G007233/1. The travels of V. Turunen were financed by the Magnus Ehrnrooth Foundation, by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters, and by the Finnish Cultural Foundation. Finally, our loving thanks go to our families for all the encouragement and understanding that we received while working on this monograph. March 2009 Michael Ruzhansky, London Ville Turunen, Helsinki