Yiheng Dong: On Fourier Restriction Estimates

Abstract

In Euclidean space \(\mathbb{R}^n\), the aim of the Fourier restriction problem is to determine the range of exponents p, q, in which one can meaningfully restrict the Fourier transform \(\hat f\) of a \(L^p\) function f to a subset S of \(\mathbb{R}^n\). The Fourier restriction problem is of central impor­tance in Fourier analysis, and is closely connected to some other fields such as partial differential equations. In this talk we will review some progress on the restriction problem and the methods involved. We will present the theorem of Tomas-­Stein and the principle of stationary phase, as well as sketch L. Guth and H. Wang’s proofs of extension estimates based on wave packet decomposition and polynomial partitioning.