数学科学学院学术报告

--- 分析、偏微分方程与动力系统讨论班(2025春季第1讲)

Sharp local Lp estimates for the Hermite eigenfunctions

王兴（湖南大学）

时间：2025年03月10日（周一）下午15:30-16:30

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https://meeting.tencent.com/p/4823449963

#腾讯会议：482-344-9963 会议密码：123456

摘要: We investigate the concentration of eigenfunctions for the Hermite operator −∆ + |x|^2 in Rn by establishing local Lp bounds over the compact sets with arbirary dilations and translations. These new results extend the local estimates by Thangavelu [49] and improve those derived from Koch-Tataru [35], and explain the special phenomenon that the global Lp bounds decrease in p when 2 ≤ p ≤ (2 n+6)/( n +1). The key L2 -estimates show that the local probabilities decrease away from the boundary {|x| = λ}, and then they satisfy Bohr’s coorespondence principle in any dimension. The proof uses the Hermite spectral projection operator represented by Mehler’s formula for the Hermite-Sch¨ordinger propagator e^{−itH}, and the strategy developed by Thangavelu and Jeong-Lee-Ryu. We also exploit an explicit version of the stationary phase lemma and H¨ormander’s L2 oscillatory integral theorem. Using

Koch-Tataru’s strategy, we construct appropriate examples to illustrate the possible concentrations and show the optimality of our local estimates.

报告人简介: 王兴，湖南大学数学学院副教授。美国约翰霍普金斯大学博士学位，师从Christopher Sogge 教授.主要研究方向是流形上的调和分析及算子谱的渐近性质，Advances in Mathematics, Canadian Journal of Mathematics, Proceedings of the American Mathematical Society，Mathematical Research Letters 等学术知名期刊上发表多篇学术论文。

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