最优光滑化估计与迹定理综述（英文）

A Survey on Optimal Smoothing Estimates and Trace Theorems

本文对自由薛定谔方程解的一大类光滑化估计中最优常数和最优函数存在性问题、以及其对偶估计式的迹定理问题近年来的研究进展进行综述.在我们近期的研究成果(含部分与町原秀二和斋藤洋树合作的成果)之前,除B.Simon和渡边一雄对部分特例得到的结果之外,关于此问题的结果很有限.基于B.Walther在更一般情况下的结果、以及我们在同一研究脉络下对经典调和分析中Funk-Hecke定理的创新应用,我们现在可以对最优常数相关的大量自然问题给出解答;本综述中介绍了前述研究成果,并阐述了证明背后的主要思路.本文还提及这一问题与带漂移项的薛定谔方程中初值问题适定性相关的沟沺—竹内猜想之间的联系,并提出一些公开问题.

We survey recent progress on the problem of obtaining the optimal constant and existence of maximizers for a broad class of smoothing estimates for the free Schrodinger propagator, together with trace theorems as their dual estimates. Before our recent work （par- tially in collaboration with S. Machihara and H. Saito）, a limited collection of results were known, including certain special cases due to B. Simon and K. Watanabe. Based on a result of B. Walther in a rather general context and our own innovation in the same vein using the Funk- Hecke theorem from classical harmonic analysis, we are now able to answer a number of natural questions regarding such sharp estimates; we exhibit these in this survey article and expose the main idea behind the proofs. We will also mention a relation with the Mizohata-Takeuchi conjecture on the well-posedness of the Cauchy problem for SchrSdinger equations with drift terms and some open problems are highlighted.