A sharp regularity estimate for the Schr\"odinger propagator on the sphere

TL;DR

This paper establishes sharp regularity estimates for the Schr"odinger propagator on spheres, providing new space-time bounds and maximal operator estimates for functions on the sphere, with implications for harmonic analysis.

Contribution

The paper proves sharp regularity estimates for the Schr"odinger propagator on spheres and derives new space-time and maximal operator bounds, extending harmonic analysis techniques.

Findings

Sharp regularity condition: ng > (n-2)/4 for the propagator estimate.

Space-time estimates for Schrf6dinger propagator on L^p spaces.

Boundedness of the maximal operator for zonal functions with ng > 1/3.

Abstract

Let $\Delta_{\mathbb S^n}$ denote the Laplace-Beltrami operator on the $n$-dimensional unit sphere $\mathbb S^n$. In this paper we show that $$ \| e^{it \Delta_{\mathbb S^n}}f \|_{L^4([0, 2\pi) \times \mathbb S^n)} \leq C \| f\|_{W^{\alpha, 4} (\mathbb S^n)} $$ holds provided that $n\geq 2$, $\alpha> {(n-2)/4}.$ The range of $\alpha$ is sharp up to the endpoint. As a consequence, we obtain space-time estimates for the Schr\"odinger propagator $e^{it \Delta_{\mathbb S^n}}$ on the $L^p$ spaces for $2\leq p\leq \infty.$ We also prove that for zonal functions on ${\mathbb S}^n$, the Schr\"odinger maximal operator $\sup_{0\leq t<2\pi} |e^{it\Delta_{\mathbb S^n}} f|$ is bounded from $W^{\alpha, 2}(\mathbb S^n) $ to $L^{\frac{6n}{3n-2}}(\mathbb S^n)$ whenever $\alpha>{1/3}$.