The endpoint Stein–Tomas inequality: old and new

Abstract

The Stein–Tomas inequality from 1975 is a cornerstone of Fourier restriction theory. Despite its respectable age, it is a fertile ground for current research. This note is centered around three classical applications – to Strichartz inequalities, Salem sets and Roth’s theorem in the primes – and three recent improvements: the sharp endpoint Stein–Tomas inequality in three space dimensions, maximal and variational refinements, and the symmetric Stein–Tomas inequality with applications.