This book is devoted to the study of pseudo-differential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space. We expose the most recent developments of the theory with its applications to local solvability and semi-classical estimates for nonselfadjoint operators. The first chapter is introductory and gives a presentation of classical classes of pseudo-differential operators. The second chapter is dealing with the general notion of metrics on the phase space. We expose some elements of the so-called Wick calculus and introduce general Sobolev spaces attached to a pseudo-differential calculus. The third and last chapter, is devoted to the topic of non-selfadjoint pseudo-differential operators. After some introductory examples, we enter into the discussion of estimates with loss of one derivative, starting with the proof of local solvability with loss of one derivative under condition (P). We show that an estimate with loss of one derivative is not a consequence of condition (Psi). Finally, we give a proof of an estimate with loss of 3/2 derivatives under condition (Psi). This book is accessible to graduate students in Analysis, and provides an up-todate overview of the subject, hopefully useful to researchers in PDE and Semi-classical Analysis.
Author(s): Nicolas Lerner
Series: Pseudo-Differential Operators Theory and Applications 3
Edition: 1st Edition.
Publisher: Birkhäuser Basel
Year: 2010
Language: English
Pages: 410
Cover......Page 1
Pseudo-Differential Operators
Theory and Applications,
Vol. 3......Page 3
Metrics on the Phase Space
and Non-Selfadjoint
Pseudo-Differential Operators......Page 4
ISBN 9783764385095......Page 5
Table of Contents......Page 6
Preface......Page 10
1.1.1 Prolegomena......Page 14
1.1.2 Quantization formulas......Page 22
1.1.3 The Sm1,0 class of symbols......Page 24
1.1.4 The semi-classical calculus......Page 35
1.2.1 Introduction......Page 41
1.2.2 Inversion of (micro)elliptic operators......Page 45
1.2.3 Propagation of singularities......Page 50
1.2.4 Local solvability......Page 55
1.3.1 Singular integrals, examples......Page 64
1.3.2 Remarks on the Calder´on-Zygmund theory and classical pseudo-differential operators......Page 67
2.1.1 Symplectic algebra......Page 70
2.1.3 Quantization formulas......Page 71
2.1.4 The metaplectic group......Page 73
2.1.5 Composition formula......Page 75
2.2.1 A short review of examples of pseudo-differential calculi......Page 80
2.2.2 Slowly varying metrics on R2n......Page 81
2.2.3 The uncertainty principle for metrics......Page 85
2.2.4 Temperate metrics......Page 87
2.2.5 Admissible metric and weights......Page 89
2.2.6 The main distance function......Page 93
2.3 General principles of pseudo-differential calculus......Page 96
2.3.2 Biconfinement estimates......Page 97
2.3.3 Symbolic calculus......Page 104
2.3.4 Additional remarks......Page 107
2.4.1 Wick quantization......Page 113
2.4.2 Fock-Bargmann spaces......Page 117
2.4.3 On the composition formula for the Wick quantization......Page 119
2.5.1 L2 estimates......Page 123
2.5.2 The G˚arding inequality with gain of one derivative......Page 126
2.5.3 The Fefferman-Phong inequality......Page 128
2.5.4 Analytic functional calculus......Page 147
2.6.1 Introduction......Page 150
2.6.2 Definition of the Sobolev spaces......Page 151
2.6.3 Characterization of pseudo-differential operators......Page 153
2.6.4 One-parameter group of elliptic operators......Page 159
2.6.5 An additional hypothesis for the Wiener lemma: the geodesic temperance......Page 165
3.1.1 Examples......Page 174
3.1.2 First-bracket analysis......Page 184
3.1.3 Heuristics on condition (Ψ)......Page 187
3.2.1 Definitions and examples......Page 190
3.2.2 Condition (P)......Page 192
3.2.3 Condition (Ψ) for semi-classical families of functions......Page 194
3.2.4 Some lemmas on C3 functions......Page 203
3.2.5 Inequalities for symbols......Page 207
3.2.6 Quasi-convexity......Page 213
3.3 The necessity of condition (Ψ)......Page 216
3.4.1 Introduction......Page 218
3.4.3 Simplifications under a more stringent condition on thesymbol......Page 220
3.5.1 Local solvability under condition (P)......Page 222
3.5.2 The two-dimensional case, the oblique derivative problem......Page 229
3.5.3 Transversal sign changes......Page 233
3.5.4 Semi-global solvability under condition (P)......Page 238
3.6.1 Introduction......Page 239
3.6.2 Construction of a counterexample......Page 245
3.6.3 More on the structure of the counterexample......Page 259
3.7.1 Introduction......Page 263
3.7.2 Energy estimates......Page 264
3.7.3 From semi-classical to local estimates......Page 276
3.8.1 A (very) short historical account of solvability questions......Page 296
3.8.2 Open problems......Page 297
3.8.3 Pseudo-spectrum and solvability......Page 298
4.1.1 Basics......Page 300
4.1.2 The logarithm of a non-singular symmetric matrix......Page 302
4.1.3 Fourier transform of Gaussian functions......Page 304
4.1.4 Some standard examples of Fourier transform......Page 308
4.1.5 The Hardy Operator......Page 312
4.2.1 On simultaneous diagonalization of quadratic forms......Page 313
4.2.2 Some remarks on commutative algebra......Page 314
4.3.1 On the Fa`a di Bruno formula......Page 316
4.3.2 On Leibniz formulas......Page 318
4.3.3 On Sobolev norms......Page 319
4.3.4 On partitions of unity......Page 321
4.3.5 On non-negative functions......Page 323
4.3.6 From discrete sums to finite sums......Page 330
4.3.7 On families of rapidly decreasing functions......Page 332
4.3.8 Abstract lemma for the propagation of singularities......Page 335
4.4.1 The symplectic structure of the phase space......Page 337
4.4.2 The metaplectic group......Page 347
4.4.3 A remark on the Feynman quantization......Page 350
4.4.4 Positive quadratic forms in a symplectic vector space......Page 351
4.5.1 Symplectic manifolds......Page 357
4.5.2 Normal forms of functions......Page 358
4.6 Composing a large number of symbols......Page 359
4.7.1 A selfadjoint operator......Page 369
4.7.2 Cotlar’s lemma......Page 370
4.7.3 Semi-classical Fourier integral operators......Page 374
4.8 On the Sj¨ostrand algebra......Page 379
4.9.1 Properties of some metrics......Page 380
4.9.2 Proof of Lemma 3.2.12 on the proper class......Page 381
4.9.3 More elements of Wick calculus......Page 383
4.9.4 Some lemmas on symbolic calculus......Page 387
4.9.5 The Beals-Fefferman reduction......Page 389
4.9.6 On tensor products of homogeneous functions......Page 391
4.9.7 On the composition of some symbols......Page 392
Bibliography......Page 396
Index......Page 408
