A decoupling proof of the Tomas restriction theorem

TL;DR

This paper presents a new proof of the Fourier restriction theorem for the paraboloid in ^n using the Bourgain-Demeter decoupling theorem, simplifying the -based restriction problem via -decoupling and - techniques.

Contribution

It introduces a decoupling-based proof of the Tomas restriction theorem, providing a more natural approach and an -removal technique for global estimates.

Findings

Decoupling theorem implies a local extension estimate with -loss.

Application of -removal techniques yields a global restriction estimate.

The proof offers a more natural and streamlined approach to restriction theorems.

Abstract

We give a new proof of a classic Fourier restriction theorem for the truncated paraboloid in $\mathbb{R}^n$ based on the $l^2$ decoupling theorem of Bourgain-Demeter. Focusing on the extension formulation of the restriction problem (dual to the original restriction formulation), we find that the $l^2$ decoupling theorem directly implies a local variant of the desired extension estimate incurring an $\varepsilon$-loss. To upgrade this result to the desired global extension estimate, we employ some $\varepsilon$-removal techniques first introduced by Tao. By adhering to the extension formulation, we obtain a more natural proof of the required $\varepsilon$-removal result.