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 which 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 which he gave at Princeton while I was his graduate student.