报告题目：Estimates for oscillatory integrals with damping factors

报告人：Sanghyuk Lee教授，首尔大学

报告时间：2025年10月21日（周二）15:45

报告地点：20-202

报告摘要：Let $\sigma$ be the surface measure on a smooth hypersurface $\mathcal H \subset \mathbb{R}^{d+1}$. A fundamental subject in harmonic analysis is to determine the decay of $\widehat{\sigma}$. For nondegenerate $\mathcal H$, the stationary phase method yields the optimal decay, while sharp bounds in the degenerate case are known only in limited situations. In this talk, we are concerned with the oscillatory estimate

$$|(\kappa^{1/2}\sigma)^\wedge(\xi)| \le C|\xi|^{-d/2},$$

for convex analytic surfaces $\mathcal H$, where $\kappa$ is the Gaussian curvature. The damping factor $\kappa^{1/2}$ is expected to  recover the optimal decay, as suggested by the stationary phase expansion, but the work of Cowling–Disney–Mauceri–Müller shows that such bounds fail in general for $d \ge 5$ even when the surface is convex and analytic. However, it has remained open whether the estimate holds in lower dimensions $2\le d\le 4$. We establish it for $d=2,3$, and with a logarithmic loss for $d=4$. Our approach is inspired by the stationary set method of Basu–Guo–Zhang–Zorin-Kranich. We also discuss applications to convolution, maximal, and adjoint restriction operators. This is joint work with Sewook Oh.

报告人简介：Sanghyuk Lee，首尔大学教授，韩国调和分析和色散偏微分方程领域的领军人物，国际著名的调和分析专家。在包括Invent. Math., J.Eur.Math.Soc., J. Math. Pures Appl., Math.Ann.等在内的众多著名数学学术期刊发表近百篇研究成果。

邀请人：流体与色散方程的分析创新团队