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Conical refraction and higher microlocalization

Author: Otto Liess
Publisher: Berlin ; New York : Springer-Verlag, ©1993.
Series: Lecture notes in mathematics (Springer-Verlag), 1555.
Edition/Format:   Print book : EnglishView all editions and formats
Database:WorldCat
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A monograph on higher analytic microlocalization and its application to problems of the propagation of singularities. The results presented refer to solutions of linear partial differential operators  Read more...

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Additional Physical Format: Online version:
Liess, Otto, 1945-
Conical refraction and higher microlocalization.
Berlin ; New York : Springer-Verlag, ©1993
(OCoLC)624430622
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Otto Liess
ISBN: 3540571051 9783540571056 0387571051 9780387571058
OCLC Number: 28421368
Description: x, 389 pages ; 24 cm.
Contents: 1. Introduction --
1.1. Statement of the main results --
1.2. An heuristic justification --
1.3. Discussion of theorem 1.1.6 --
2. Higher order wave front sets --
2.1. Higher order wave front sets --
2.2. [actual symbol not reproducible]-microlocalization --
2.3. Splitting and approximation of entire functions --
2.4. Higher microlocalization and duality --
2.5. Plurisubharmonic functions related to propagation phenomena --
2.6. An improvement of proposition 2.5.10 --
2.7. Proof of theorem 2.1.12 --
2.8. Proof of theorem 2.1.13 --
2.9. Proof of proposition 2.1.15 --
A.1 Propagation of analyticity for partially holomorphic functions --
A.2 Proof of lemma 2.5.1 --
A.3 Proof of proposition 2.1.7 --
3. Pseudodifferential operators --
3.1. Polyhomogeneous structures and polycones --
3.2. Polyhomogeneity and polyhomogeneous symbols --
3.3. Symbol classes and pseudodifferential operators in general [actual symbol not reproducible]-microlocalization --
3.4. Pseudodifferential operators in k-microlocalization --
3.5. Polyhomogeneity and symbols --
3.6. Finite sums of polyhomogeneous functions and principal parts. Relative Poisson brackets --
3.7. Successive localizations --
3.8. Regularity theorems I --
3.9. Regularity theorems II. Proof of theorem 1.1.6 --
3.10. Improvements in the case of constant coefficient operators --
3.11. Proof of proposition 3.10.3 --
3.12. Proof of the propositions 1.1.9 and 1.1.12 --
3.13. Remarks on localization of polynomials. Proof of proposition 1.1.13 --
4. Bi-symplectic geometry and multihomogeneous maps --
4.1. Polyhomogeneous changes of coordinates in R[superscript n] --
4.2. Polyhomogeneous changes of coordinates in R[superscript 2n] --
4.3. Bihomogeneous approximations of homogeneous maps --
4.4. The bihomogeneous structure of N[Sigma] --
4.5. The canonical foliation and the relative tangent space of N[Sigma] --
4.6. The bisymplectic structure of N[Sigma] --
4.7. The relative bicharacteristics foliation of [Lambda] --
4.8. Phase functions and bihomogeneous approximations of canonical maps --
4.9. Construction of the phase --
4.10. Reduction of p[subscript m] to model form, inequalities and a regularity theorem --
4.11. Canonical transformations and estimates for symbols --
5. Fourier Integral Operators --
5.1. Classical F.I.O.'s which leave [actual symbol not reproducible]= 0 invariant --
5.2. Invariant definition of [actual symbol not reproducible] --
5.3. F.I.O. associated with phase functions which live on bineighborhoods --
5.4. Phase functions associated with Lipschitzian weight functions --
5.5. Composition of F.I.O. --
5.6. Invariant meaning and proof of theorem 1.1.3 --
6. Conical refraction, hyperbolicity and slowness surfaces --
6.1. Influence domains and bicharacteristics --
6.2. The canonical form of operators when [actual symbol not reproducible] is second order and hyperbolic --
6.3. The case when p[subscript m, k] is hyperbolic --
6.4. Singular points on surfaces --
6.5. Remarks on the velocity, the slowness and the wave surface --
6.6. Singular points on the slowness surface and conical refraction --
6.7. Conical refraction in free space for the system of crystal optics --
6.8. Singular points on the slowness surface of the system of elasticity for cubic crystals --
7. Propagation of regularity up to the boundary --
7.1. Interior regularity when the traces are smooth --
7.2. Extension across hypersurfaces of solutions of constant coefficient microlocally hyperbolic operators --
7.3. The wave front set of a restriction --
7.4. Estimates for the Fourier transform of some surface densities --
7.5. Boundary values and partial regularity --
7.6. Regularity up to the boundary --
7.7. Proof of lemma 7.6.5 --
7.8. Extension of solutions from the interior --
8. Some results on transmission problems --
8.1. Formulation of the problem --
8.2. Regularity of traces as a consequence of interior regularity, I --
8.3. Regularity of traces as a consequence of interior regularity, II --
8.4. Incoming and outgoing bicharacteristics and propagation cones --
8.5. Conical refraction in transmission problems --
8.6. Conical reflection at the boundary --
8.7. The case of crystal optics --
8.8. Other phenomena: External conical refraction --
9. Partial analyticity, higher microlocalization and sheaves --
9.1. The [actual symbol not reproducible]-class of a partially analytic distribution --
9.2. The first two wave front sets and partial analyticity --
9.3. Proof of proposition 9.1.2 --
9.4. Review of hyperfunctions --
9.5. Hyperfunctions and real analytic functionals --
9.6. Partially analytic distributions and infinite order operators --
9.7. Proof of proposition 9.1.3 --
9.8. The sheaf of partially analytic microfunctions --
9.9. Second microlocalization and sheaves --
9.10. The Fourier transform of measures on C[superscript n] with supports concentrated on polycones --
9.11. Higher order wave front sets and boundary values of holomorphic functions.
Series Title: Lecture notes in mathematics (Springer-Verlag), 1555.
Responsibility: Otto Liess.
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