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

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.