On extremizing sequences for the adjoint restriction inequality on the cone

TL;DR

This paper investigates the behavior of extremizing sequences for the adjoint Fourier restriction inequality on the cone in three-dimensional space, proving their precompactness and convergence after applying symmetries, building on known extremizer forms.

Contribution

It establishes the precompactness and convergence of nonnegative extremizing sequences for the cone restriction inequality, utilizing symmetry transformations and known extremizer structures.

Findings

Nonnegative extremizing sequences are precompact after symmetry adjustments.

Extremizing sequences converge when the exact extremizer form is used.

The results rely on the known explicit form of extremizers by Carneiro.

Abstract

It is known that extremizers for the $L^2$ to $L^6$ adjoint Fourier restriction inequality on the cone in $\mathbb{R}^3$ exist. Here we show that nonnegative extremizing sequences are precompact, after the application of symmetries of the cone. If we use the knowledge of the exact form of the extremizers, as found by Carneiro, then we can show that nonnegative extremizing sequences converge, after the application of symmetries.