Mathematical Sciences: Geometric Nonlinear Harmonic Analysis

数学科学：几何非线性调和分析

• 批准号: 9400230
• 负责人: Stephen Semmes
• 金额: --
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-01 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400230&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Nonlinear; Harmonic

9400230 Semmes The proposed research involves a combination of harmonic analysis and geometric measure theory that is somewhat complicated. Conventionally one likes to ask questions about the relationship between the geometry of a set in a Euclidean space and the analysis of functions and linear operators associated to this set. One can also consider the geometry and complexity of sets in Euclidean spaces for their own sake, and look for methods to detect when a set has some nice interesting structure, e.g., when it can be parametrized in a nice way with estimates. For this latter endeavor harmonic analysis has a slightly indirect relevance, by providing guidance for what "interesting structure" might mean, as well as useful techniques. However, many of the traditional tools of harmonic analysis do not apply in this inherently nonlinear setting, and in fact there are results in the geometric setting whose counterparts in classical analysis are false. The work of Semmes involves the interplay between measure theory, geometry and analysis. Given a set of points in space, one can ask whether it can be measured. In the case of a one dimensional set, this is the question of whether it has finite length. This can be a very subtle question in analysis. Classical answers to this question often involve boundedness of certain operations on spaces of functions on the set. In this project answers to the question of "measurability" will be sought where the classical approaches cannot work. This will involve replacing the classical operations by more complicated nonlinear operations. ***

所提出的研究涉及到谐波分析和几何测量理论的结合，有些复杂。通常人们喜欢问欧几里得空间中一个集合的几何形状和与这个集合相关的函数和线性算子的分析之间的关系。人们也可以考虑欧几里德空间中集合的几何和复杂性，并寻找方法来检测一个集合何时具有一些有趣的结构，例如，何时可以用估计的方式很好地参数化。对于后一种努力，谐波分析具有轻微的间接关联，通过提供“有趣的结构”可能意味着什么的指导，以及有用的技术。然而，许多传统的谐波分析工具并不适用于这种固有的非线性环境，事实上，在几何环境中，经典分析中的对应结果是错误的。Semmes的工作涉及测量理论、几何和分析之间的相互作用。给定空间中的一组点，人们可以问它是否可以测量。在一维集合的情况下，这是关于它是否有有限长度的问题。这在分析中可能是一个非常微妙的问题。这个问题的经典答案通常涉及集合上函数空间上某些操作的有界性。在这个项目中，“可测量性”问题的答案将在经典方法无法工作的地方寻求。这将涉及到用更复杂的非线性运算取代经典运算。＊＊＊