歐氏空間之傅立葉變換限制問題的探討

Abstract

本論文的目的是研究歐幾里得空間中的傅立葉限制問題及其在法爾科納距離集猜想中的應用。限制問題是調和分析領域中最知名的研究問題之一，且與其他研究領域（如偏微分方程和幾何測度論）有著重要的聯繫。在本論文中，我們主要詳細介紹已知的Tomas-Stein成果、雙線性限制估計以及Bourgain對法爾科納距離猜想的結果。

The purpose of this dissertation is to study the Fourier restriction problems in Euclidean spaces and their applications to Falconer's distance set conjecture. Restriction problems are one of the most known research problems in the area of Harmonic analysis and have been found to have important connections to other research fields such as partial differential equations and geometric measure theory. In this dissertation, we mainly introduce in details the known Tomas-Stein results, bilinear restriction estimates and Bourgain's work on Falconer distance conjecture.