The Analysis of Linear Partial Differential Operators III: Pseudo-Differential OperatorsFrom the reviews: "Volumes III and IV complete L. Hörmander'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. Brüning in Zentralblatt MATH, 1987. |
Contents
Second Order Elliptic Operators Summary | 3 |
171 Interior Regularity and Local Existence Theorems | 4 |
172 Unique Continuation Theorems | 9 |
173 The Dirichlet Problem | 24 |
174 The Hadamard Parametrix Construction | 30 |
175 Asymptotic Properties of Eigenvalues and Eigenfunctions | 42 |
Notes | 61 |
PseudoDifferential Operators | 63 |
Notes | 346 |
Some Classes of MicroHypoelliptic Operators | 348 |
221 Operators With PseudoDifferential Parametrix | 349 |
222 Generalized Kolmogorov Equations | 353 |
223 Melins Inequality | 359 |
224 Hypoellipticity with Loss of One Derivative | 366 |
Notes | 383 |
The Strictly Hyperbolic Cauchy Problem | 385 |
181 The Basic Calculus | 65 |
182 Conormal Distributions | 96 |
183 Totally Characteristic Operators | 112 |
184 Gauss Transforms Revisited | 141 |
185 The Weyl Calculus | 150 |
186 Estimates of PseudoDifferential Operators | 161 |
Notes | 178 |
Elliptic Operators on a Compact Manifold Without Boundary | 180 |
192 The Index of Elliptic Operators | 193 |
193 The Index Theorem in IR | 215 |
194 The Lefschetz Formula | 222 |
195 Miscellaneous Remarks on Ellipticity | 225 |
Notes | 229 |
Boundary Problems for Elliptic Differential Operators | 231 |
201 Elliptic Boundary Problems | 232 |
202 Preliminaries on Ordinary Differential Operators | 251 |
203 The Index for Elliptic Boundary Problems | 255 |
204 NonElliptic Boundary Problems | 264 |
Notes | 266 |
Symplectic Geometry | 268 |
211 The Basic Structure | 269 |
212 Submanifolds of a Symplectic Manifold | 283 |
213 Normal Forms of Functions | 296 |
214 Folds and Glancing Hypersurfaces | 303 |
215 Symplectic Equivalence of Quadratic Forms | 321 |
216 The Lagrangian Grassmannian | 328 |
232 Operators of Higher Order | 390 |
233 Necessary Conditions for Correctness of the Cauchy Problem | 400 |
234 Hyperbolic Operators of Principal Type | 404 |
Notes | 414 |
The Mixed DirichletCauchy Problem for Second Order Operators | 416 |
242 Singularities in the Elliptic and Hyperbolic Regions | 423 |
243 The Generalized Bicharacteristic Flow | 430 |
244 The Diffractive Case | 443 |
245 The General Propagation of Singularities | 455 |
246 Operators Microlocally of Tricomis Type | 460 |
247 Operators Depending on Parameters | 465 |
Notes | 469 |
Appendix B Some Spaces of Distributions | 471 |
B2 Distributions in a Half Space and in a Manifold with Boundary | 478 |
Appendix C Some Tools from Differential Geometry | 485 |
C2 A Singular Differential Equation | 487 |
C3 Clean Intersections and Maps of Constant Rank | 490 |
C4 Folds and Involutions | 492 |
C5 Geodesic Normal Coordinates | 500 |
C6 The Morse Lemma with Parameters | 502 |
Notes | 504 |
Bibliography | 505 |
| 523 | |
Common terms and phrases
a₂ adjoint assume asymptotic bicharacteristic boundary problem bounded Cauchy problem choose coefficients compact set completes the proof condition conic neighborhood conormal conormal bundle constant continuous convergent Corollary defined Definition denote derivatives differential equations differential operator eigenvalues elliptic operators equal estimate finite follows from Theorem formula Fourier transform Fredholm operator function H₂ half density Hence homogeneous of degree hyperbolic hypothesis implies integral isomorphism kernel Lemma linear Math norm obtain orthogonal parametrix principal symbol proof is complete proof of Theorem properly supported Proposition prove pseudo-differential operators quadratic form replaced respect restriction right-hand side S₂ satisfying Section singularities solution subset supp symplectic coordinates symplectic form symplectic manifold symplectic vector space tangent transversal unique valid vanishes variables vector bundle vector field wave front set x₁ y₁
Popular passages
Page 519 - Schapira, P. Propagation at the boundary and reflection of analytic singularities of solutions of linear partial differential equations. Publ. RIMS Kyoto Univ.
Page 521 - Introduction to Pseudodifferential and Fourier Integral Operators, Volume 1: Pseudodifferential Operators, Volume 2: Fourier Integral Operators, by Francois Treves; Plenum Press, New York, 1980; 649 pages, $29.95 & 35.00, cloth.
Page 512 - On surface waves with finite and infinite speed of propagation, Arch. Rational Mech. Anal. 19 (1965).