The analysis of linear partial differential operators. IV: Fourier integral operators
type: Item
instance of: scholarly article
title: The analysis of linear partial differential operators. IV: Fourier integral operators
zbMATH Open document ID: 1178.35003
DOI: 10.1007/978-3-642-00136-9
author: Lars Hörmander
published in: Classics in Mathematics
publication date: 2009-03-12T00:00:00Z
review text: The fourth volume of the impressive monograph ``The Analysis of Partial Differential Operators'' by Lars Hörmander continues the detailed and unified approach of pseudo-differential and Fourier integral operators. The present book is a paperback edition of the fourth volume of this monograph. For a detailed description of the contents of the book we refer the reader to the review of the first edition [The analysis of linear partial differential operators. III: Pseudo-differential operators. Grundlehren der Mathematischen Wissenschaften, 274. Berlin etc.: Springer-Verlag (1985; Zbl 0601.35001)]. There is a longstanding connection, where symplectic geometry plays a fundamental role, between partial differential equations and geometrical and wave optics, classical and quantum mechanics. The techniques emerging from this tradition form the object of the present book: Fourier integral operators, with the study of the asymptotic properties of eigenvalues of self-adjoint elliptic operators on compact manifolds without boundary; the systematic study of the propagation of singularities, by micro-local analysis, with the Cauchy problem and pseudo-differential operators of principal type; long range scattering theory. The book has been one of the most elegant, inspiring and influential texts in Mathematics, and contains results used in many important developments in Partial Differential Equations and Mathematical Physics. Listing these results would be a daunting task. We would only mention here that results and ideas in Chapter XXVI were used in the proof of the sufficiency of the \(\Psi\)-condition for the local solvability of principal type pseudo-differential operators [see \textit{N. Dencker}, Ann. Math. (2) 163, No.~2, 405--444 (2006; Zbl 1104.35080)], the necessity of this condition having been proved by [\textit{L.~Hörmander}, Singularities in boundary value problems, Proc. NATO Adv. Study Inst., Maratea/Italy 1980, 69--96 (1981; Zbl 0459.35096)]. See the joint ``Looki…
reviewed by: Viorel Iftimie
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35Axx
Mathematics Subject Classification ID: 35Sxx
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 47G40
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53D05
zbMATH DE Number: 5528184
zbMATH Keywords: Fourier integral operators,
zbMATH Keywords: pseudo-differential operators
zbMATH Keywords: Weyl calculus
zbMATH Keywords: operators of principal type
zbMATH Keywords: local regularity
zbMATH Keywords: local solvability
zbMATH Keywords: subelliptic operators
zbMATH Keywords: Cauchy problem
zbMATH Keywords: symplectic geometry
zbMATH Keywords: long-range scattering
zbMATH Keywords: Hamiltonian flow
zbMATH Keywords: bicharacteristic
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-00136-9
OpenAlex ID: W2491079642
Wikidata QID: Q110898007
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description: scientific article; zbMATH DE number 5528184