Extremizers for the Airy-Strichartz inequality

TL;DR

This paper determines the precise threshold for the existence of optimizers in the Airy--Strichartz inequality, linking it to the sharp constant in the classical Strichartz inequality, and establishing conditions for optimizer existence.

Contribution

It explicitly identifies the compactness threshold for optimizing sequences of the Airy--Strichartz inequality relative to the sharp Strichartz constant, covering full parameter ranges.

Findings

Threshold for optimizer existence is an explicit multiple of the sharp Strichartz constant.

Optimizer existence is guaranteed when the Airy--Strichartz sharp constant is below this threshold.

Results apply to all non-endpoint cases in both diagonal and off-diagonal regimes.

Abstract

We identify the compactness threshold for optimizing sequences of the Airy-- Strichartz inequality as an explicit multiple of the sharp constant in the Strichartz inequality. In particular, if the sharp constant in the Airy--Strichartz inequality is strictly smaller than this multiple of the sharp constant in the Strichartz inequality, then there is an optimizer for the former inequality. Our result is valid for the full range of Airy--Strichartz inequalities (except the endpoints) both in the diagonal and off-diagonal cases.