Not long ago I had the pleasure of reviewing J. J. Duistermaat and J. A. C. Kolk’s fine book, Distributions: Theory and Applications. Now I have occasion to talk about another Duistermaat effort, Fourier Integral Operators, appearing as a re-issue in Birkhäuser’s “Modern Classics” series. The late Johannes Jisse Duistermaat, who generally went by Hans, passed away a little over a year ago in Utrecht, Holland, at only 67 years of age. He was a very highly respected mathematician whose marvelous expository abilities were clearly show-cased in his books. The book under review certainly qualifies under this description; it also attests to the breadth of Duistermaat’s interests. Indeed, the obituary Univ. Utrecht published last year includes the following: “[Duistermaat] was fond of mathematical problems that required the sharp tools of analysis combined with the deep insights of geometry. In the field of differential equations he was a leading figure in the world.”
Duistermaat’s Fourier Integral Operators had its genesis in a course the author taught at Nijmegen in 1970. The lecture notes have an interesting history. The Courant Institute first distributed them, but they eventually grew too rare: enter Birkhäuser Verlag in 1995 to launch a “nicely TeXed version of these notes with some minor additions … and corrections,” and now, fifteen years later, we are the recipients of Birkhäuser’s wise move to re-issue the book as a Modern Classic.
And rightly so. For the properly prepared and properly disposed mathematical audience Fourier Integral Operators is a must. As Duistermaat’s Introduction already shows, this is, in the true sense of the word, hard analysis. Even the 0th chapter bespeaks specialization: the author starts with abbreviated (if standard) notations for partial differential operators in the Schwartz space over Rn, notes that Fourier transformation gives an automorphism of this space and then launches a discussion of PDE and pseudodifferential operators in this setting, culminating with a brief literature review of sorts surrounding the work of Lax, Hörmander, and Maslov. And we’re only on p. 7.
In other words, the reader had better gird his loins for what lies ahead. In fact, a very solid background in analysis is nigh on indispensable for mining a maximum yield out of the dense 140 pages of the book. But one of the treasures in Fourier Integral Operators is quite unusual and amplifies the worth of this (already very valuable) book considerably: Duistermaat finishes his second chapter with a discussion of oscillatory integrals with nondegenerate phase functions and then proceeds to consider symplectic differential geometry, Lagrangian manifolds, and even the attendant calculus of variations: this is in fact central to the book, even as the closing parts of Fourier Integral Operators are concerned with such hard-analysis mainstays as the global theory of Fourier integral operators, the Cauchy problem, and oscillatory asymptotic solutions to a certain class of PDE having classical roots.
The earlier focus on symplectic geometry is of independent value, however, because this subject straddles a half dozen areas of contemporary mathematics (including number theory and physics), making Fourier Integral Operators something of a double threat: certain fellow-travelers for whom hard-analysis is, well, too hard to take, will want to cover these parts of the book under review, given what is presented there in so compact and careful a fashion.
Fourier Integral Operators is not easy going: there is no chattiness, there are no exercise sets, and, again, there is a lot of hard analysis (and even a bit of soft analysis) in the book’s 140 pages. But it is a very important book on a subject that is both deep and broad.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.
Preface.- 0. Introduction.- 1. Preliminaries.- 1.1 Distribution densities on manifolds.- 1.2 The method of stationary phase.- 1.3 The wave front set of a distribution.- 2. Local Theory of Fourier Integrals.- 2.1 Symbols.- 2.2 Distributions defined by oscillatory integrals.- 2.3 Oscillatory integrals with nondegenerate phase functions.- 2.4 Fourier integral operators (local theory).- 2.5 Pseudodifferential operators in Rn.- 3. Symplectic Differential Geometry.- 3.1 Vector fields.- 3.2 Differential forms.- 3.3 The canonical 1- and 2-form T* (X).- 3.4 Symplectic vector spaces.- 3.5 Symplectic differential geometry.- 3.6 Lagrangian manifolds.- 3.7 Conic Lagrangian manifolds.- 3.8 Classical mechanics and variational calculus.- 4. Global Theory of Fourier Integral Operators.- 4.1 Invariant definition of the principal symbol.- 4.2 Global theory of Fourier integral operators.- 4.3 Products with vanishing principal symbol.- 4.4 L2-continuity.- 5. Applications.- 5.1 The Cauchy problem for strictly hyperbolic differential operators with C-infinity coefficients.- 5.2 Oscillatory asymptotic solutions. Caustics.- References.