A remark on restriction theorems and their application to Strichartz estimates

TL;DR

This paper introduces an elementary method for proving restriction theorems on specific surfaces, leading to simplified proofs of Strichartz estimates for PDEs like the wave and Euler equations.

Contribution

It provides a novel, straightforward approach to restriction theorems that bypasses the Tomas-Stein theorem, enabling easier derivation of key PDE estimates.

Findings

Simplified proofs of Strichartz estimates for the wave equation.

New restriction theorem applications for surfaces where Tomas-Stein does not apply.

Alternative derivation of estimates related to Euler equations.

Abstract

We present an elementary approach to prove restriction theorems for particular surfaces for which the Tomas-Stein theorem does not apply, which in turn provide short proofs for well-known Strichartz estimates for associated PDEs. The method consists in applying simple restriction theorems to the level sets of the surfaces studied (for instance, spheres in the case of the cone) and then integrating the inequalities over all level sets. This allows for a different proof of sharp estimates for the wave equation, and of some estimates related to the Euler equations in the rotational framework.