The Analysis of Linear PD Operators. III, Pseudo-Differential Operators

Lars Hörmander3540499377, 9783540499374

“Volumes III and IV complete L. H?rmander’s treatise on linear partial differential equations. They constitute the most complete and up-to-date account of this subject, by the author who has dominated it and made the most significant contributions in the last decades…..It is a superb book, which must be present in every mathematical library, and an indispensable tool for all – young and old – interested in the theory of partial differential operators.” L. Boutet de Monvel in Bulletin of the American Mathematical Society, 1987.

“This treatise is outstanding in every respect and must be counted among the great books in mathematics. It is certainly no easy reading (…) but a careful study is extremely rewarding for its wealth of ideas and techniques and the beauty of presentation.” J. Br?ning in Zentralblatt MATH, 1987.

Table of contents :
Cover……Page 1
Series……Page 2
Title page……Page 3
Date-line……Page 4
Series of 1994 Edition……Page 5
Title page of 1994 Edition……Page 6
Date-line of 1994 Edition……Page 7
Preface……Page 8
Contents……Page 9
Introduction……Page 12
Summary……Page 14
17.1. Interior Regularity and Local Existence Theorems……Page 15
17.2. Unique Continuation Theorems……Page 20
17.3. The Dirichlet Problem……Page 35
17.4. The Hadamard Parametrix Construction……Page 41
17.5. Asymptotic Properties of Eigenvalues and Eigenfunctions……Page 53
Notes……Page 72
Summary……Page 74
18.1. The Basic Calculus……Page 76
18.2. Conormal Distributions……Page 107
18.3. Totally Characteristic Operators……Page 123
18.4. Gauss Transforms Revisited……Page 152
18.5. The Weyl Calculus……Page 161
18.6. Estimates of Pseudo-Differential Operators……Page 172
Notes……Page 189
19.1. Abstract Fredholm Theory……Page 191
19.2. The Index of Elliptic Operators……Page 204
19.3. The Index Theorem in $mathbb{R}^n$……Page 226
19.4. The Lefschetz Formula……Page 233
19.5. Miscellaneous Remarks on Ellipticity……Page 236
Notes……Page 240
Summary……Page 242
20.1. Elliptic Boundary Problems……Page 243
20.2. Preliminaries on Ordinary Differential Operators……Page 262
20.3. The Index for Elliptic Boundary Problems……Page 266
20.4. Non-Elliptic Boundary Problems……Page 275
Notes……Page 277
Summary……Page 279
21.1. The Basic Structure……Page 280
21.2. Submanifolds of a Sympletic Manifold……Page 294
21.3. Normal Forms of Functions……Page 307
21.4. Folds and Glancing Hypersurfaces……Page 314
21.5. Symplectic Equivalence of Quadratic Forms……Page 332
21.6. The Lagrangian Grassmannian……Page 339
Notes……Page 357
Summary……Page 359
22.1. Operators with Pseudo-Differential Parametrix……Page 360
22.2. Generalized Kolmogorov Equations……Page 364
22.3. Melin’s Inequality……Page 370
22.4. Hypoellipticity with Loss of One Derivative……Page 377
Notes……Page 394
23.1. First Order Operators……Page 396
23.2. Operators of Higher Order……Page 401
23.3. Necessary Conditions for Correctness of the Cauchy Problem……Page 411
23.4. Hyperbolic Operators of Principal Type……Page 415
Notes……Page 425
24.1. Energy Estimates and Existence Theorems in the Hyperbolic Case……Page 427
24.2. Singularities in the Elliptic and Hyperbolic Regions……Page 434
24.3. The Generalized Bicharacteristic Flow……Page 441
24.4. The Diffractive Case……Page 454
24.5. The General Propagation of Singularities……Page 466
24.6. Operators Microlocally of Tricomi’s Type……Page 471
24.7. Operators Depending on Parameters……Page 476
Notes……Page 480
B.1. Distributions in $mathbb{R}^n$ and in an Open Manifold……Page 482
B.2. Distributions in a Half Space and in a Manifold with Boundary……Page 489
C.1. The Frobenius Theorem and Foliations……Page 496
C.2. A Singular Differential Equation……Page 498
C.3. Clean Intersections and Maps of Constant Rank……Page 501
C.4. Folds and Involutions……Page 503
C.5. Geodesic Normal Coordinates……Page 511
C.6. The Morse Lemma with Parameters……Page 513
Notes……Page 515
Bibliography……Page 516
Index……Page 534
Index of Notation……Page 536