Extremizers for Adjoint Restriction to a Pair of Reflected Paraboloids

TL;DR

This paper investigates extremizers for the adjoint restriction inequality related to reflected paraboloids, establishing their existence in all dimensions and analyzing the precompactness of extremizing sequences under certain conditions.

Contribution

It proves the existence of extremizers for the adjoint restriction inequality on reflected paraboloids in all dimensions and verifies a key inequality in low dimensions.

Findings

Extremizers exist in all dimensions.

Extremizing sequences are precompact modulo symmetries under a certain inequality.

The key inequality is verified for dimensions 1 and 2.

Abstract

We consider the adjoint restriction inequality associated to the hypersurface $\{(\tau, \xi) : \tau = \pm|\xi|^2, \;\xi \in \mathbb{R}^d\}$ at the Stein-Tomas exponent. Extremizers exist in all dimensions and extremizing sequences are precompact modulo symmetries conditional on a certain inequality, which we verify in the case $d \in \{1,2\}$.