On extremizers for Strichartz estimates for higher order Schr\"odinger equations

TL;DR

This paper investigates extremizers for Strichartz estimates related to higher order Schrödinger equations, computing optimal constants, proving non-existence of extremizers, and resolving key open questions in the field.

Contribution

It introduces a new comparison principle for convolutions of singular measures and precisely determines operator norms for certain higher order Schrödinger equations.

Findings

Optimal constants for the extension inequality are computed.

Extremizers do not exist for the studied class of equations.

The work resolves a key open problem regarding extremizer existence.

Abstract

For an appropriate class of convex functions $\phi$, we study the Fourier extension operator on the surface $\{(y, |y|^2+\phi(y)):y\in\mathbb{R}^2\}$ equipped with projection measure. For the corresponding extension inequality, we compute optimal constants and prove that extremizers do not exist. The main tool is a new comparison principle for convolutions of certain singular measures that holds in all dimensions. Using tools of concentration-compactness flavor, we further investigate the behavior of general extremizing sequences. Our work is directly related to the study of extremizers and optimal constants for Strichartz estimates of certain higher order Schr\"odinger equations. In particular, we resolve a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schr\"odinger equations, and compute the corresponding operator norms exactly where only lower bounds were previously known.