Strichartz estimates for the Schr\"odinger equation on Zoll manifolds

TL;DR

This paper establishes optimal space-time Strichartz estimates for the Schrödinger equation on Zoll manifolds, including spheres, by leveraging spectral arithmetic properties and bilinear oscillatory integral techniques.

Contribution

It introduces a novel approach connecting spectral properties of Zoll manifolds to Strichartz estimates, extending results to all $q \, \geq 2$.

Findings

Optimal $L^q_{t,x}$ estimates for Schrödinger solutions on Zoll manifolds.

Extension of Strichartz estimates to the standard sphere $S^d$.

Use of spectral arithmetic properties and bilinear oscillatory integrals.

Abstract

We obtain optimal space-time estimates in $L^q_{t,x}$ spaces for all $q\ge 2$ for solutions to the Schr\"odinger equation on Zoll manifolds, including, in particular, the standard round sphere $S^d$. The proof relies on the arithmetic properties of the spectrum of the Laplacian on Zoll manifolds, as well as bilinear oscillatory integral estimates, which allow us to relate the problem to Strichartz estimate on one-dimensional tori.