限制性猜想是现代调和分析的核心问题之一,是联系几何测度论中Kakeya猜想、Bocher-Riesz猜想、波动方程解的局部光滑猜想的桥梁.主题是研究光滑曲面的支撑测度的Fourier变换在无穷远处的衰减和曲面的几何性质(包括曲率和大小尺度)之间的联系.著名的 Stein-Tomas限制性定理在PDE中对应着Strichartz估计,已成为研究非线性色散方程的基本工具. 本项目致力于发展波包分解、尺度归纳及堆垒组合方法, 研究自然出现各种变系数的限制性估计（具位势的Laplace算子、光滑紧流形的自伴拟微分算子、平坦环上Laplace算子等对应的限制性问题)及相关核心问题.进而，将这些限制性估计与profiles 分解、集中紧致与刚性方法相结合,着力研究具位势或光滑紧流形上的非线性色散方程的动力学行为、锥奇性空间波动方程的奇性传播、具位势或光滑紧流形线性色散方程解的极大估计与点态收敛等.

Restriction conjecture is one of the core problems of modern harmonic analysis. It is a bridge connecting well-known problems such as Kakeya's conjecture, Bocher-Riesz's conjecture and local smoothness conjecture on solution of wave equation. The subject is to study the relation between the decay of infinity on the Fourier transform of the support measure of a smooth surface and the geometric properties of the surface (including curvature and size scale). The well-known Strichartz estimates corresponds to PDEs version of Stein-Tomas restriction thorem , and it has become the basic tool for studying nonlinear dispersion equations. he project is devoted to the development of wave packet decomposition, scale induction and addition combination argument methods to study natural variable coefficient restriction estimaties (restriction estimaties associated with: Laplace operator with potential, self-adjoint differential operators on smooth compact manifolds, Laplace operators on flat torus , etc.) and related core issues. Furthermore, combining these restriction estimates with the profiles decomposition, the compactness and rigid method, we try our best to study the dynamic behavior of the nonlinear dispersion equation with potential or on smooth compact manifolds (we will explore in depth the influence of the geometry and the manifold geometry on the dynamics behavior), the singularity propagation on solution of wave equation in the conic singular space, the maximal estimate of the solution of the linear dispersion equation with potential or the linear dispersion equation in smooth compact manifolds , and the related problem on pointwise convergence,etc..