The Analysis of Linear Partial Differential Operators. IV, Fourier Integral Operators

Lars Hormander9780387138299, 0387138293

From the reviews: These two volumes (III & IV) complete L. Hoermander’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. Bull. AMS 16,1 (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. ZBfM 612 (1987)

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Table of contents :
Title page……Page 1
Title……Page 2
Date-line……Page 3
Preface to Volumes III and IV……Page 4
Contents……Page 5
Introduction……Page 8
Summary……Page 10
25.1. Lagrangian Distributions……Page 11
25.2. The Calculus of Fourier Integral Operators……Page 24
25.3. Special Cases of the Calculus, and $L^2$ Continuity……Page 31
25.4. Distributions Associated with Positive Lagrangian Ideals……Page 42
25.5. Fourier Integral Operators with Complex Phase……Page 50
Notes……Page 59
Summary……Page 61
26.1. Operators with Real Principal Symbols……Page 64
26.2. The Complex Involutive Case……Page 80
26.3. The Symplectic Case……Page 88
26.4. Solvability and Condition ($Psi$)……Page 98
26.5. Geometrical Aspects of Condition ($P$)……Page 117
26.6. The Singularities in $N_{11}$……Page 124
26.7. Degenerate Cauchy-Riemann Operators……Page 130
26.8. The Nirenberg-Treves Estimate……Page 141
26.9. The Singularities in $N^e_2$ and in $N^e_{12}$……Page 144
26.10. The Singularities on One Dimensional Bicharacteristics……Page 156
26.11. A Semi-Global Existence Theorem……Page 168
Notes……Page 170
27.1. Definitions and Main Results……Page 172
27.2. The Taylor Expansion of the Symbol……Page 178
27.3. Subelliptic Operators Satisfying ($P$)……Page 185
27.4. Local Properties of the Symbol……Page 190
27.5. Local Subelliptic Estimates……Page 209
27.6. Global Subelliptic Estimates……Page 219
Notes……Page 226
28.1. Calderon’s Uniqueness Theorem……Page 227
28.2. General Carleman Estimates……Page 241
28.3. Uniqueness Under Convexity Conditions……Page 246
28.4. Second Order Operators of Real Principal Type……Page 249
Notes……Page 255
29.1. The Spectral Measure and its Fourier Transform……Page 256
29.2. The Case of a Periodic Hamilton Flow……Page 270
29.3. The Weyl Formula for the Dirichlet Problem……Page 278
Notes……Page 281
Summary……Page 283
30.1. Admissible Perturbations……Page 284
30.2. The Boundary Value of the Resolvent, and the Point Spectrum……Page 288
30.3. The Hamilton Flow……Page 303
30.4. Modified Wave Operators……Page 315
30.5. Distorted Fourier Transforms and Asymptotic Completeness……Page 321
Notes……Page 337
Bibliography……Page 339
Index……Page 357
Index of Notation……Page 359