Interpolation Theorems in Harmonic Analysis

TL;DR

This thesis explores key interpolation theorems in harmonic analysis, providing background, proofs, and applications to PDEs, Sobolev spaces, and Fourier integral operators, highlighting their theoretical significance and practical utility.

Contribution

It offers a comprehensive exposition of four major interpolation theorems, including proofs and applications, enriching understanding in harmonic analysis and PDE theory.

Findings

Proved the Riesz-Thorin interpolation theorem and its extension by Stein.

Applied interpolation theorems to PDEs, Sobolev spaces, and Fourier integral operators.

Provided detailed background and examples to facilitate understanding of complex interpolation methods.

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.