A sharp regularity estimate for the Schrödinger propagator on the sphere

2020/12/11 by Chen, Xianghong, Duong, Xuan Thinh, Lee, Sanghyuk +1 · 1 citation
#35B45 #35J0 #42B37 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.06313

Abstract

Let Δ_{\mathbb Sⁿ} denote the Laplace-Beltrami operator on the n-dimensional unit sphere \mathbb Sⁿ. In this paper we show that ‖ e^it Δ_{\mathbb Sⁿ}f ‖_{L⁴([0, 2π) × \mathbb Sⁿ)} ≤ C ‖ f‖_{W^{α, 4} (\mathbb Sⁿ)} holds provided that n≥ 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 Sⁿ} on the L^p spaces for 2≤ p≤ ∞. We also prove that for zonal functions on \mathbb Sⁿ, the Schrödinger maximal operator sup_{0≤ t<2π} |e^itΔ_{\mathbb Sⁿ} f| is bounded from W^{α, 2}(\mathbb Sⁿ) to L^(6n)/(3n-2)(\mathbb Sⁿ) whenever α>1/3.

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• Regularity and pointwise convergence for dispersive equations on Riemannian symmetric spaces of compact type

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