A sharp regularity estimate for the Schrödinger propagator on the sphere

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 $$\Vert e^{it{\Delta }_{{\mathbb{S}}^n}}f{\Vert }_{L^4([0,2\pi )\times {\mathbb{S}}^n)}\le C\Vert f{\Vert }_{W^{\alpha ,4}({\mathbb{S}}^n)}$$ holds provided that $n\ge 2$, $\alpha >(n-2)/4.$ The range of $\alpha$ is sharp up to the endpoint. As a consequence, we obtain space-time estimates for the Schrödinger propagator $e^{it{\Delta }_{{\mathbb{S}}^n}}$ on the $L^p$ spaces for $2\le p\le \infty .$ We also prove that for zonal functions on ${\mathbb{S}}^n$, the Schrödinger maximal operator ${\sup }_{0\le t<2\pi }|e^{it{\Delta }_{{\mathbb{S}}^n}}f|$ is bounded from $W^{\alpha ,2}({\mathbb{S}}^n)$ to $L^{\frac{6n}{3n-2}}({\mathbb{S}}^n)$ whenever $\alpha >1/3$.