0 - Background

Summary

The purpose of this chapter and the next is to present the background material that will be needed. The topics are standard and a more thorough treatment can be found in many excellent sources, such as Stein [2] and Stein and Weiss [1] for the first half and Hörmander [7, Vol. 1] for the second.

We start out by rapidly going over basic results from real analysis, including standard theorems concerning the Fourier transform in ℝⁿ and Caldéron-Zygmund theory. We then apply this to prove the Hardy-Littlewood-Sobolev inequality. This theorem on fractional integration will be used throughout and we shall also present a simple argument showing how the n-dimensional theorem follows from the original one-dimensional inequality of Hardy and Littlewood. This type of argument will be used again and again. Finally, in the last two sections we give the definition of the wave front set of a distribution and compute the wave front sets of distributions which are given by oscillatory integrals. This will be our first encounter with the cotangent bundle and, as the monograph progresses, this will play an increasingly important role.