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Strichartz estimates for the Schrödinger equation on Zoll manifolds

Xiaoqi Huang, Christopher D. Sogge

TL;DR

This work proves optimal space-time Strichartz estimates for the Schrödinger equation on Zoll manifolds for all $q\ge 2$, with the data-to-solution loss $s$ exceeding the exponent $\mu(q)$ determined by a combination of Sobolev and spectral-cluster constraints. The authors combine the arithmetic structure of the Zoll spectrum with bilinear oscillatory integral techniques, reducing dispersive control to Strichartz-type bounds on a one-dimensional torus and employing a dyadic, microlocal analysis to treat delicate low-exponent regimes. They resolve the full range in all dimensions, including the challenging $d=2$, $4<q<6$ case, and establish sharpness (up to $\varepsilon$) via spectral and zonal-function examples. The results extend the understanding of dispersive phenomena on compact curved spaces and have implications for Strichartz theory on positive-curvature space forms.

Abstract

We obtain optimal space-time estimates in $L^q_{t,x}$ spaces for all $q\ge 2$ for solutions to the Schrödinger 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.

Strichartz estimates for the Schrödinger equation on Zoll manifolds

TL;DR

This work proves optimal space-time Strichartz estimates for the Schrödinger equation on Zoll manifolds for all , with the data-to-solution loss exceeding the exponent determined by a combination of Sobolev and spectral-cluster constraints. The authors combine the arithmetic structure of the Zoll spectrum with bilinear oscillatory integral techniques, reducing dispersive control to Strichartz-type bounds on a one-dimensional torus and employing a dyadic, microlocal analysis to treat delicate low-exponent regimes. They resolve the full range in all dimensions, including the challenging , case, and establish sharpness (up to ) via spectral and zonal-function examples. The results extend the understanding of dispersive phenomena on compact curved spaces and have implications for Strichartz theory on positive-curvature space forms.

Abstract

We obtain optimal space-time estimates in spaces for all for solutions to the Schrödinger equation on Zoll manifolds, including, in particular, the standard round sphere . 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.
Paper Structure (4 sections, 7 theorems, 138 equations)

This paper contains 4 sections, 7 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}.
  3. Proof of Theorem \ref{['main']} when $4<q<6$ and $d=2$.
  4. Sharpness of Theorem \ref{['main']}

Key Result

Theorem 1

Let $(M, g)$ be a smooth Zoll manifold of dimension $d\ge 2$. Then for all $q\ge 2$, the solution $u$ of 00.2 satisfies where $\mu(q)=\max\{\frac{d-1}{2}(\frac{1}{2}-\frac{1}{q}), \frac{d-1}{2}-\frac{d}{q}, \frac{d}{2}-\frac{d+2}{q}\}$.

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 1
  • Lemma 2
  • proof : Proof of Lemma \ref{['diag1']}
  • Proposition 3
  • Lemma 4
  • proof : Proof of Lemma \ref{['leelemma']}
  • Proposition 5