Some Variants of the Kakeya Problem

(1)这个项目将专注于经典分析中的某些几何问题，这些问题起源于著名的Kakeya问题：找到在每个方向上包含线段的d维空间子集的最小维度。私家侦探特别感兴趣。是从下面的集合的Hausdorff维数的估计，这些集合与某些线，平面或球体的集合（或者，在更高的维度中，与某些k平面或超球体的集合）有很大的交集。这项工作的一个主题将是探索一种特定方法的范围和局限性，由于p.i.，用于处理这些问题的研究中出现的平均算子。(2)这个项目关注的是高维空间的某些复杂子集的结构。对这些集合的研究，虽然具有令人信服的内在兴趣，但应该被正确地视为对我们目前对某些称为平均算子的对象的理解的测试。这里相关的平均算子有两种：球面平均算子和Radon变换。球面平均算子是我们理解波在流体介质（如海洋或大气）中传播的基础。Radon变换及其相关变换在诸如医学成像领域中出现的断层摄影问题中是必不可少的。

(1) This project will focus on certain geometric questions from classical analysis which have their origins in the well-known Kakeya problem: find the minimum dimension of a subset of d-dimensional space which contains a line segment in each direction. Of particular interest to the p.i. are estimates from below for the Hausdorff dimension of sets which have large intersections with certain collections of lines, planes, or spheres (or, in higher dimensions, with certain collections of k-planes or hyperspheres). A theme of this work will be to explore the scope and limitations of a particular method, due to the p.i., for treating the averaging operators which arise in the study of these problems. (2) This project is concerned with the structure of certain complicated subsets of higher dimensional spaces. The study of these sets, while of compelling intrinsic interest, should properly be viewed as a test of our current understanding of certain objects called averaging operators. The relevant averaging operators here are of two kinds: spherical averaging operators and Radon transforms. Spherical averaging operators are basic to our understanding of the propagation of waves through fluid media such as the ocean or the atmosphere. The Radon transform and its relatives are essential, for example, in problems of tomography such as those which arise in the field of medical imaging.