Factorisation in Restriction theory and near extremisers

TL;DR

This paper presents a new, simpler method using induction-on-scales to analyze Fourier restriction problems on quadratic surfaces, constructing near extremisers with significant mass, offering an alternative to classical factorisation techniques.

Contribution

It introduces an induction-on-scales approach for restriction theory, avoiding traditional factorisation methods, and constructs near extremisers with large mass for quadratic surfaces.

Findings

Applicable to paraboloid and hyperbolic paraboloid surfaces

Provides an alternative to Maurey-Nikishin-Pisier factorisation

Constructs near extremisers with significant mass

Abstract

We give an alternative argument to the application of the so-called Maurey- Nikishin-Pisier factorisation in Fourier restriction theory. Based on an induction-on-scales argument, our comparably simple method applies to any compact quadratic surface, in particular compact parts of the paraboloid and the hyperbolic paraboloid. This is achieved by constructing near extremisers with big "mass", which itself might be of interest.