Maximizers for the Stein-Tomas inequality

TL;DR

This paper establishes a criterion for the precompactness of optimizing sequences in the Stein-Tomas inequality and links the existence of optimizers to a conjecture in Strichartz inequalities, applicable across all dimensions.

Contribution

It provides a necessary and sufficient condition for precompactness and connects the existence of optimizers to a well-known conjecture, advancing understanding of the inequality's extremizers.

Findings

Precompactness criterion for optimizing sequences.

Conditional existence of optimizers based on a conjecture.

Applicable in all dimensions.

Abstract

We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein-Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein-Tomas inequality. Our result is valid in any dimension.