论文标题
Schrödinger传播器在球体上的尖锐规律性估计值
A sharp regularity estimate for the Schrödinger propagator on the sphere
论文作者
论文摘要
令$δ_ {\ mathbb s^n} $表示$ n $ dimensional单位球上的laplace-beltrami操作员$ \ mathbb s^n $。在本文中，我们表明$$ \ | e^{ITδ_ {\ Mathbb s^n}} f \ | _ {l^4（[[0，2π）\ times \ times \ mathbb s^n）} \ leq c \ | f \ | _ {w^{α，4}（\ mathbb s^n）} $$保留，只要$ n \ geq 2 $，$α> {（n-2）/4}。$ $α$的范围是尖锐的端点。结果，我们在$ l^p $ spaces上以$ 2 \ leq p \ leq p \ leq \infty。$在$ l^p $ space上获得Schrödinger繁殖者$ e^{uq p $ spaces上的Δ_ {\ mathbb s^n}} $的时空估计。 $ \ sup_ {0 \ leq t <2π} | e^{utδ_ {\ mathbb s^n}} f | $从$ w^{α，2}（\ mathbb s^n）$到$ w^{α，2} $ $α> {1/3} $。
Let $Δ_{\mathbb S^n}$ denote the Laplace-Beltrami operator on the $n$-dimensional unit sphere $\mathbb S^n$. In this paper we show that $$ \| e^{it Δ_{\mathbb S^n}}f \|_{L^4([0, 2π) \times \mathbb S^n)} \leq C \| f\|_{W^{α, 4} (\mathbb S^n)} $$ holds provided that $n\geq 2$, $α> {(n-2)/4}.$ The range of $α$ is sharp up to the endpoint. As a consequence, we obtain space-time estimates for the Schrödinger propagator $e^{it Δ_{\mathbb S^n}}$ on the $L^p$ spaces for $2\leq p\leq \infty.$ We also prove that for zonal functions on ${\mathbb S}^n$, the Schrödinger maximal operator $\sup_{0\leq t<2π} |e^{itΔ_{\mathbb S^n}} f|$ is bounded from $W^{α, 2}(\mathbb S^n) $ to $L^{\frac{6n}{3n-2}}(\mathbb S^n)$ whenever $α>{1/3}$.
