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

这个项目的目的是研究在调和分析中起核心作用的一些算子的性质。这些是傅里叶变换、平均算子、受限X射线变换和极大算子的限制。统一上述研究的是，在每种情况下，都存在其几何属性(例如曲率)决定相应算子的映射属性的基本曲线或曲面。在许多情况下，目标是对大类标的品种保持一致的估计也是很自然的。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，我们将需要进一步理解一些问题，如关于牛顿多面体的傅里叶变换的衰减性质，涉及称为仿射弧长的重要量的几何不等式，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.