Maximal restriction estimates and the maximal function of the Fourier transform

TL;DR

This paper establishes a maximal restriction inequality for the Fourier transform, advancing understanding of Fourier analysis by answering an open question and introducing a novel linearisation technique.

Contribution

It provides the first proof of a maximal restriction inequality for the Fourier transform and introduces a new trick for linearising maximal functions.

Findings

Proves a maximal restriction inequality for the Fourier transform

Introduces a new linearisation trick for maximal functions

Discusses potential for stronger high-dimensional results

Abstract

We prove a maximal restriction inequality for the Fourier transform, providing an answer to a question left open by M\"uller, Ricci and Wright. Our methods are similar to the ones in their article, with the addition of a suitable trick to help us linearise our maximal function. In the end, we comment on how to use the same trick in combination with Vitturi's approach to obtain a stronger high-dimensional result.