Extremizability of Fourier restriction to the paraboloid

TL;DR

This paper proves the existence and precompactness of extremizers for all nonendpoint Fourier restriction inequalities on the paraboloid in certain dimensions, extending previous results and linking to the restriction conjecture.

Contribution

It establishes the existence and compactness of extremizers for all nonendpoint restriction inequalities on the paraboloid, generalizing prior work for the case q=2.

Findings

Existence of extremizers for all nonendpoint inequalities.

Precompactness of extremizing sequences modulo symmetries.

Conditional results linked to the restriction conjecture.

Abstract

In this article, we prove that all global, nonendpoint Fourier restriction inequalities for the paraboloid in $\mathbb R^{1+d}$ have extremizers and that $L^p$-normalized extremizing sequences are precompact modulo symmetries. This result had previously been established for the case $q=2$. In the range where the boundedness of the restriction operator is still an open question, our result is conditional on improvements toward the restriction conjecture.