Problems in Euclidean harmonic analysis related to the geometry of curves and surfaces

这个项目的目的是研究在谐波分析中起核心作用的一些算子的性质。这些是傅里叶变换的限制，平均算子，受限的x射线变换和极大算子。统一上述研究的是，在每种情况下，存在一些潜在的曲线或曲面，其几何性质（例如曲率）决定了相应算子的映射性质。在许多情况下，力求对大类基础品种的估计是一致的也是很自然的。这些统一的估计在某种意义上是最优的。为了实现我们的目标，我们需要进一步理解一些问题，比如傅里叶变换与牛顿多面体的衰减特性，涉及一个重要的量称为仿射弧长的几何不等式，以及它们与某些组合技术的相互作用。

The purpose of this project is to investigate the properties of a number of operators that play a central role in harmonic analysis. These are the restriction of the Fourier transform, averaging operators, the restricted X-ray transform and maximal operators. What unifies the study of the above is the presence, in each case, of some underlying curve or surface whose geometric properties (e.g. the curvature) determine the mapping properties of the corresponding operators. In many cases it is also natural to aim for estimates that are uniform over large classes of the underlying varieties. These uniform estimates are optimal in a certain sense. In order to achieve our goals, we will need to further our understanding of issues such as the decay properties of the Fourier transform in relation to Newton polyhedral, geometric inequalities involving an important quantity called the affine arc length and their interplay with certain combinatorial techniques.