Lerner Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators

1. Auflage 2011
ISBN: 978-3-7643-8510-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 3, 397 Seiten

Reihe: Pseudo-Differential Operators

This book is devoted to the study of pseudo-di?erential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space. We have tried here to expose the most recent developments of the theory with its applications to local solvability and semi-classical estimates for non-selfadjoint operators. The?rstchapter,Basic Notions of Phase Space Analysis,isintroductoryand gives a presentation of very classical classes of pseudo-di?erential operators, along with some basic properties. As an illustration of the power of these methods, we give a proof of propagation of singularities for real-principal type operators (using aprioriestimates,andnotFourierintegraloperators),andweintroducethereader to local solvability problems. That chapter should be useful for a reader, say at the graduate level in analysis, eager to learn some basics on pseudo-di?erential operators. The second chapter, Metrics on the Phase Space begins with a review of symplectic algebra, Wigner functions, quantization formulas, metaplectic group and is intended to set the basic study of the phase space. We move forward to the more general setting of metrics on the phase space, following essentially the basic assumptions of L. H¨ ormander (Chapter 18 in the book [73]) on this topic.

Autoren/Hrsg.

Lerner, Nicolas

Inhaltsverzeichnis

1;Title Page ;3
2;Copyright Page ;4
3;Table of Contents ;5
4;Preface;9
5;Chapter 1 Basic Notions of Phase Space Analysis;12
5.1;1.1 Introduction to pseudo-differential operators;12
5.1.1;1.1.1 Prolegomena;12
5.1.2;1.1.2 Quantization formulas;20
5.1.3;1.1.3 The Sm1,0 class of symbols;22
5.1.4;1.1.4 The semi-classical calculus;33
5.2;1.2 Pseudo-differential operators on an open subset of Rn;39
5.2.1;1.2.1 Introduction;39
5.2.2;1.2.2 Inversion of (micro)elliptic operators;43
5.2.3;1.2.3 Propagation of singularities;48
5.2.4;1.2.4 Local solvability;53
5.3;1.3 Pseudo-differential operators in harmonic analysis;62
5.3.1;1.3.1 Singular integrals, examples;62
5.3.2;1.3.2 Remarks on the Calder´on-Zygmund theory and classical pseudo-differential operators;65
6;Chapter 2 Metrics on the Phase Space;67
6.1;2.1 The structure of the phase space;67
6.1.1;2.1.1 Symplectic algebra;67
6.1.2;2.1.2 The Wigner function;68
6.1.3;2.1.3 Quantization formulas;68
6.1.4;2.1.4 The metaplectic group;70
6.1.5;2.1.5 Composition formula;72
6.2;2.2 Admissible metrics;77
6.2.1;2.2.1 A short review of examples of pseudo-differential calculi;77
6.2.2;2.2.2 Slowly varying metrics on R2n;78
6.2.3;2.2.3 The uncertainty principle for metrics;82
6.2.4;2.2.4 Temperate metrics;84
6.2.5;2.2.5 Admissible metric and weights;86
6.2.6;2.2.6 The main distance function;90
6.3;2.3 General principles of pseudo-differential calculus;93
6.3.1;2.3.1 Confinement estimates;94
6.3.2;2.3.2 Biconfinement estimates;94
6.3.3;2.3.3 Symbolic calculus;101
6.3.4;2.3.4 Additional remarks;104
6.3.5;2.3.5 Changing the quantization;110
6.4;2.4 The Wick calculus of pseudo-differential operators;110
6.4.1;2.4.1 Wick quantization;110
6.4.2;2.4.2 Fock-Bargmann spaces;114
6.4.3;2.4.3 On the composition formula for the Wick quantization;116
6.5;2.5 Basic estimates for pseudo-differential operators;120
6.5.1;2.5.1 L2 estimates;120
6.5.2;2.5.2 The G°arding inequality with gain of one derivative;123
6.5.3;2.5.3 The Fefferman-Phong inequality;125
6.5.4;2.5.4 Analytic functional calculus;144
6.6;2.6 Sobolev spaces attached to a pseudo-differential calculus;147
6.6.1;2.6.1 Introduction;147
6.6.2;2.6.2 Definition of the Sobolev spaces;148
6.6.3;2.6.3 Characterization of pseudo-differential operators;150
6.6.4;2.6.4 One-parameter group of elliptic operators;156
6.6.5;2.6.5 An additional hypothesis for the Wiener lemma: the geodesic temperance;162
7;Chapter 3 Estimates for Non-Selfadjoint Operators;171
7.1;3.1 Introduction;171
7.1.1;3.1.1 Examples;171
7.1.2;3.1.2 First-bracket analysis;181
7.1.3;3.1.3 Heuristics on condition (.);184
7.2;3.2 The geometry of condition (.);187
7.2.1;3.2.1 Definitions and examples;187
7.2.2;3.2.2 Condition (P);189
7.2.3;3.2.3 Condition (.) for semi-classical families of functions;191
7.2.4;3.2.4 Some lemmas on C3 functions;200
7.2.5;3.2.5 Inequalities for symbols;204
7.2.6;3.2.6 Quasi-convexity;210
7.3;3.3 The necessity of condition (.);213
7.4;3.4 Estimates with loss of k/k + 1 derivative;215
7.4.1;3.4.1 Introduction;215
7.4.2;3.4.2 The main result on subellipticity;217
7.4.3;3.4.3 Simplifications under a more stringent condition on thesymbol;217
7.5;3.5 Estimates with loss of one derivative;219
7.5.1;3.5.1 Local solvability under condition (P);219
7.5.2;3.5.2 The two-dimensional case, the oblique derivative problem;226
7.5.3;3.5.3 Transversal sign changes;230
7.5.4;3.5.4 Semi-global solvability under condition (P);235
7.6;3.6 Condition (.) does not imply solvability with loss of one derivative;236
7.6.1;3.6.1 Introduction;236
7.6.2;3.6.2 Construction of a counterexample;242
7.6.3;3.6.3 More on the structure of the counterexample;256
7.7;3.7 Condition (.) does imply solvability with loss of 3/2 derivatives;260
7.7.1;3.7.1 Introduction;260
7.7.2;3.7.2 Energy estimates;261
7.7.3;3.7.3 From semi-classical to local estimates;273
7.8;3.8 Concluding remarks;293
7.8.1;3.8.1 A (very) short historical account of solvability questions;293
7.8.2;3.8.2 Open problems;294
7.8.3;3.8.3 Pseudo-spectrum and solvability;295
8;Chapter 4 Appendix;297
8.1;4.1 Some elements of Fourier analysis;297
8.1.1;4.1.1 Basics;297
8.1.2;4.1.2 The logarithm of a non-singular symmetric matrix;299
8.1.3;4.1.3 Fourier transform of Gaussian functions;301
8.1.4;4.1.4 Some standard examples of Fourier transform;305
8.1.5;4.1.5 The Hardy Operator;309
8.2;4.2 Some remarks on algebra;310
8.2.1;4.2.1 On simultaneous diagonalization of quadratic forms;310
8.2.2;4.2.2 Some remarks on commutative algebra;311
8.3;4.3 Lemmas of classical analysis;313
8.3.1;4.3.1 On the Fa`a di Bruno formula;313
8.3.2;4.3.2 On Leibniz formulas;315
8.3.3;4.3.3 On Sobolev norms;316
8.3.4;4.3.4 On partitions of unity;318
8.3.5;4.3.5 On non-negative functions;320
8.3.6;4.3.6 From discrete sums to finite sums;327
8.3.7;4.3.7 On families of rapidly decreasing functions;329
8.3.8;4.3.8 Abstract lemma for the propagation of singularities;332
8.4;4.4 On the symplectic and metaplectic groups;334
8.4.1;4.4.1 The symplectic structure of the phase space;334
8.4.2;4.4.2 The metaplectic group;344
8.4.3;4.4.3 A remark on the Feynman quantization;347
8.4.4;4.4.4 Positive quadratic forms in a symplectic vector space;348
8.5;4.5 Symplectic geometry;354
8.5.1;4.5.1 Symplectic manifolds;354
8.5.2;4.5.2 Normal forms of functions;355
8.6;4.6 Composing a large number of symbols;356
8.7;4.7 A few elements of operator theory;366
8.7.1;4.7.1 A selfadjoint operator;366
8.7.2;4.7.2 Cotlar’s lemma;367
8.7.3;4.7.3 Semi-classical Fourier integral operators;371
8.8;4.8 On the Sj¨ostrand algebra;376
8.9;4.9 More on symbolic calculus;377
8.9.1;4.9.1 Properties of some metrics;377
8.9.2;4.9.2 Proof of Lemma 3.2.12 on the proper class;378
8.9.3;4.9.3 More elements of Wick calculus;380
8.9.4;4.9.4 Some lemmas on symbolic calculus;384
8.9.5;4.9.5 The Beals-Fefferman reduction;386
8.9.6;4.9.6 On tensor products of homogeneous functions;388
8.9.7;4.9.7 On the composition of some symbols;389
9;Bibliography;393
10;Index;405