Except for minor modifications, this monograph represents the lecture notes of a course I gave at UCLA during the winter and spring quarters of 1991. My purpose in the course was to present the necessary background material and to show how ideas from the theory of Fourier integral operators can be useful for studying basic topics in classical analysis, such as oscillatory integrals and maximal functions. The link between the theory of Fourier integral operators and classical analysis is of course not new, since one of the early goals of microlocal analysis was to provide variable coefficient versions of the Fourier transform. However, the primary goal of this subject was to develop tools for the study of partial differential equations and, to some extent, only recently have many classical analysts realized its utility in their subject. In these notes I attempted to stress the unity between these two subjects and only presented the material from microlocal analysis that would be needed for the later applications in Fourier analysis. I did not intend for this course to serve as an introduction to microlocal analysis. For this the reader should be referred to the excellent treatises of Hörmander [5], [7] and Treves [1].

In addition to these sources, I also borrowed heavily from Stein [4]. His work represents lecture notes based on a course that he gave at Princeton while I was his graduate student. As the reader can certainly tell, this course influenced me quite a bit and I am happy to acknowledge my indebtedness. My presentation of the overlapping material is very similar to his, except that I chose to present the material in the chapter on oscillatory integrals more geometrically, using the cotangent bundle. This turns out to be useful in dealing with Fourier analysis on manifolds and it also helps to motivate some results concerning Fourier integral operators, in particular the local smoothing estimates at the end of the monograph.

Roughly speaking, the material is organized as follows. The first two chapters present background material on Fourier analysis and stationary phase that will be used throughout. The next chapter deals with nonhomogeneous oscillatory integrals. It contains the L2 restriction theorem for the Fourier transform, estimates for Riesz means in Rn, and Bourgain's circular maximal theorem.

In the twenty plus years since I wrote the first edition of this book there have been many important developments in harmonic and microlocal analysis. For instance, very recently there has been seminal work on multilinear methods through the breakthrough of Bennett, Carbery and Tao [1], the related oscillatory integral estimates of Bourgain and Guth [1] and the decoupling estimates of Bourgain and Demeter [1]. These works and others have led to many applications, including the proof of sharp local smoothing estimates. I have chosen not to include these, partly because they represent a fast moving target, but moreover because they would have added considerably to the book's length. My goal for the first as well as the current edition was to provide a self-contained but by no means exhaustive introduction to harmonic and microlocal analysis.

To this end, I added two new chapters. One, which ties in with the earlier material on Fourier integral operators, presents Hörmander's theory of propagation of singularities and uses this to prove the Duistermaat–Guillemin theorem, which is a triumph of microlocal methods, saying that for generic manifolds one can obtain an improvement in the error term for the Weyl law. This chapter includes material that perhaps should have been included in the first edition and augments the microlocal analysis portion of the book. I also added a new chapter on what could now be considered classical harmonic analysis. It concerns results related to the Kakeya conjecture, including the “bush” method of Bourgain and the “hairbrush” method ofWolff for obtaining Kakeya and Nikodym maximal estimates, as well as the relationship between the Fourier restriction problem and the Kakeya conjecture. Most of the results presented in this chapter were being developed as the first edition of my book was being completed.

I also added a bit of new material to the chapters in the first edition, including Bourgain's counterexample to certain oscillatory integral estimates, which showed that Stein's oscillatory integral theorem was sharp, and I also made a few corrections to the earlier text. This turned out to be a more difficult undertaking than I anticipated due to the fact that the first edition of the book was written in the age of floppy disks and although I had saved the output files on several of them, I did not do the same for the source files for the first edition.