Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators (eBook)

Title Page 3
Copyright Page 4
Table of Contents 5
Preface 9
Chapter 1 Basic Notions of Phase Space Analysis 12
1.1 Introduction to pseudo-differential operators 12
1.1.1 Prolegomena 12
1.1.2 Quantization formulas 20
1.1.3 The Sm1,0 class of symbols 22
1.1.4 The semi-classical calculus 33
1.2 Pseudo-differential operators on an open subset of Rn 39
1.2.1 Introduction 39
1.2.2 Inversion of (micro)elliptic operators 43
1.2.3 Propagation of singularities 48
1.2.4 Local solvability 53
1.3 Pseudo-differential operators in harmonic analysis 62
1.3.1 Singular integrals, examples 62
1.3.2 Remarks on the Calder´on-Zygmund theory and classical pseudo-differential operators 65
Chapter 2 Metrics on the Phase Space 67
2.1 The structure of the phase space 67
2.1.1 Symplectic algebra 67
2.1.2 The Wigner function 68
2.1.3 Quantization formulas 68
2.1.4 The metaplectic group 70
2.1.5 Composition formula 72
2.2 Admissible metrics 77
2.2.1 A short review of examples of pseudo-differential calculi 77
2.2.2 Slowly varying metrics on R2n 78
2.2.3 The uncertainty principle for metrics 82
2.2.4 Temperate metrics 84
2.2.5 Admissible metric and weights 86
2.2.6 The main distance function 90
2.3 General principles of pseudo-differential calculus 93
2.3.1 Confinement estimates 94
2.3.2 Biconfinement estimates 94
2.3.3 Symbolic calculus 101
2.3.4 Additional remarks 104
2.3.5 Changing the quantization 110
2.4 The Wick calculus of pseudo-differential operators 110
2.4.1 Wick quantization 110
2.4.2 Fock-Bargmann spaces 114
2.4.3 On the composition formula for the Wick quantization 116
2.5 Basic estimates for pseudo-differential operators 120
2.5.1 L2 estimates 120
2.5.2 The G°arding inequality with gain of one derivative 123
2.5.3 The Fefferman-Phong inequality 125
2.5.4 Analytic functional calculus 144
2.6 Sobolev spaces attached to a pseudo-differential calculus 147
2.6.1 Introduction 147
2.6.2 Definition of the Sobolev spaces 148
2.6.3 Characterization of pseudo-differential operators 150
2.6.4 One-parameter group of elliptic operators 156
2.6.5 An additional hypothesis for the Wiener lemma: the geodesic temperance 162
Chapter 3 Estimates for Non-Selfadjoint Operators 171
3.1 Introduction 171
3.1.1 Examples 171
3.1.2 First-bracket analysis 181
3.1.3 Heuristics on condition (.) 184
3.2 The geometry of condition (.) 187
3.2.1 Definitions and examples 187
3.2.2 Condition (P) 189
3.2.3 Condition (.) for semi-classical families of functions 191
3.2.4 Some lemmas on C3 functions 200
3.2.5 Inequalities for symbols 204
3.2.6 Quasi-convexity 210
3.3 The necessity of condition (.) 213
3.4 Estimates with loss of k/k + 1 derivative 215
3.4.1 Introduction 215
3.4.2 The main result on subellipticity 217
3.4.3 Simplifications under a more stringent condition on thesymbol 217
3.5 Estimates with loss of one derivative 219
3.5.1 Local solvability under condition (P) 219
3.5.2 The two-dimensional case, the oblique derivative problem 226
3.5.3 Transversal sign changes 230
3.5.4 Semi-global solvability under condition (P) 235
3.6 Condition (.) does not imply solvability with loss of one derivative 236
3.6.1 Introduction 236
3.6.2 Construction of a counterexample 242
3.6.3 More on the structure of the counterexample 256
3.7 Condition (.) does imply solvability with loss of 3/2 derivatives 260
3.7.1 Introduction 260
3.7.2 Energy estimates 261
3.7.3 From semi-classical to local estimates 273
3.8 Concluding remarks 293
3.8.1 A (very) short historical account of solvability questions 293
3.8.2 Open problems 294
3.8.3 Pseudo-spectrum and solvability 295
Chapter 4 Appendix 297
4.1 Some elements of Fourier analysis 297
4.1.1 Basics 297
4.1.2 The logarithm of a non-singular symmetric matrix 299
4.1.3 Fourier transform of Gaussian functions 301
4.1.4 Some standard examples of Fourier transform 305
4.1.5 The Hardy Operator 309
4.2 Some remarks on algebra 310
4.2.1 On simultaneous diagonalization of quadratic forms 310
4.2.2 Some remarks on commutative algebra 311
4.3 Lemmas of classical analysis 313
4.3.1 On the Fa`a di Bruno formula 313
4.3.2 On Leibniz formulas 315
4.3.3 On Sobolev norms 316
4.3.4 On partitions of unity 318
4.3.5 On non-negative functions 320
4.3.6 From discrete sums to finite sums 327
4.3.7 On families of rapidly decreasing functions 329
4.3.8 Abstract lemma for the propagation of singularities 332
4.4 On the symplectic and metaplectic groups 334
4.4.1 The symplectic structure of the phase space 334
4.4.2 The metaplectic group 344
4.4.3 A remark on the Feynman quantization 347
4.4.4 Positive quadratic forms in a symplectic vector space 348
4.5 Symplectic geometry 354
4.5.1 Symplectic manifolds 354
4.5.2 Normal forms of functions 355
4.6 Composing a large number of symbols 356
4.7 A few elements of operator theory 366
4.7.1 A selfadjoint operator 366
4.7.2 Cotlar’s lemma 367
4.7.3 Semi-classical Fourier integral operators 371
4.8 On the Sj¨ostrand algebra 376
4.9 More on symbolic calculus 377
4.9.1 Properties of some metrics 377
4.9.2 Proof of Lemma 3.2.12 on the proper class 378
4.9.3 More elements of Wick calculus 380
4.9.4 Some lemmas on symbolic calculus 384
4.9.5 The Beals-Fefferman reduction 386
4.9.6 On tensor products of homogeneous functions 388
4.9.7 On the composition of some symbols 389
Bibliography 393
Index 405