Interpolation Theorems in Harmonic Analysis

Abstract

This expository thesis contains a study of four interpolation theorems, the requisite background material, and a few applications. The materials introduced in the first three sections of Chapter 1 are used to motivate and prove the Riesz-Thorin interpolation theorem and its extension by Stein, both of which are presented in the fourth section. Chapter 2 revolves around Calder\'{o}n's complex method of interpolation and the interpolation theorem of Fefferman and Stein, with the material in between providing the necessary examples and tools. The two theorems are then applied to a brief study of linear partial differential equations, Sobolev spaces, and Fourier integral operators, presented in the last section of the second chapter.