Sharp Strichartz inequalities for fractional and higher order Schr\"odinger equations

TL;DR

This paper studies sharp Fourier extension inequalities related to fractional and higher order Schrödinger equations, establishing existence of extremizers for certain p-values, analyzing their decay properties, and exploring applicability to related curves.

Contribution

It proves the existence of extremizers for Strichartz inequalities in specific regimes, resolves a prior dichotomy, and introduces a geometric comparison principle for convolutions of singular measures.

Findings

Extremizers exist for 1<p<p_0, with p_0>4.

Extremizers exhibit fast L^2-decay and entire Fourier transforms.

The methods extend to certain related planar curves.

Abstract

We investigate a class of sharp Fourier extension inequalities on the planar curves $s=|y|^p$, $p>1$. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if $1<p<p_0$, for some $p_0>4$. In particular, this resolves the dichotomy of Jiang, Pausader & Shao concerning the existence of extremizers for the Strichartz inequality for the fourth order Schr\"odinger equation in one spatial dimension. One of our tools is a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$, developed in a companion paper. We further show that any extremizer exhibits fast $L^2$-decay in physical space, and so its Fourier transform can be extended to an entire function on the whole complex plane. Finally, we investigate the extent to which our methods apply to the case of the planar curves $s=y|y|^{p-1}$, $p>1$.