Extremizers for Adjoint Restriction to Pairs of Translated Paraboloids

TL;DR

This paper investigates the extremizers of an adjoint restriction inequality for a union of two translated paraboloids, proving their non-existence and characterizing extremizing sequences via the paraboloid case.

Contribution

It establishes the non-existence of extremizers for the union of two translated paraboloids and characterizes extremizing sequences through known paraboloid extremizers.

Findings

Extremizers do not exist for the union of two translated paraboloids.

Extremizing sequences are characterized in terms of paraboloid extremizers.

Provides a complete description of extremizing sequences for the inequality.

Abstract

Consider the adjoint restriction inequality associated with the hypersurface $\{ (\tau, \xi) \in \mathbb{R}^{d+1} : \tau = |\xi|^2 \} \cup \{(\tau, \xi) \in \mathbb{R}^{d+1} : \tau - \tau_0 = |\xi - \xi_0|^2\}$ for any $(\tau_0, \xi_0) \neq 0$. We prove that extremizers do not exist for this inequality and fully characterize extremizing sequences in terms of extremizers for the adjoint restriction inequality for the paraboloid.