Problems in harmonic analysis

谐波分析中的问题

• 批准号: 0245401
• 负责人: Steven Hofmann
• 金额: $ 30.06万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245401&HistoricalAwards=false
• 关键词: Problems; harmonic; analysis

AbstractS. HofmannUniversity of MissouriDMS 245401 In this project, I shall consider problems in harmonic analysis arising in geometric analysis and geometric measure theory, and in the theory of elliptic and parabolic partial differential equationsand operators. These questions are interesting because of theirapplicability to real world phenomena (see below), but also because of the deep connections among them. The main directions of the proposed research include the following: 1. To develop and apply ``T1/Tb" (i.e., Carleson measure) criteria for the solvability of boundary problems for divergence form elliptic equations and systems. 2. To treat various problems in the theory of uniformly rectifiable sets, in the applications of this theory to elliptic and parabolic PDE, and in the theory of quasiconformal mappings. 3. To obtain sharp average decay estimates for Fourier transforms, and to apply these to concrete problems including lattice point problems andthe Falconer distance problem. As mentioned above, I propose to work on problems in the area of harmonic analysis and its application to, and interaction with, geometric measure theory and the theory of elliptic and parabolicpartial differential equations. Roughly speaking, in harmonic analysis one investigates properties offunctions and ``operators" (i.e., mappings which transform one functioninto another) by decomposing them into smaller, constituent pieces,which are easier to understand, and then reassembling the pieces. Thename itself arose by analogy to the decomposition of a musical soundinto its various frequency components, or ``harmonics".geometric measure theory involves the study of the relationship betweengeometric properties of sets, and their ``measures" (the latter aregeneralizations of the notions of length, area, and volume). Partialdifferential equations of elliptic and of parabolic type describe a wide variety of phenoma in the real world, including electrostatics, certain fluid flows and elastic deformations, and various diffusionprocessessuch as the conduction of heat, the flow of ground water, certainphenomena arising in the mathematical theory of population biology, and the pricing of options in financial markets. In the last decade the interplay between these different subfields of mathematics has turned out to be afertile ground for investigation, with much exciting work remaining to be done.

摘要。 MISSOURIDMS 245401的Hofmannuniversity在该项目中，我将考虑在几何分析和几何测量理论以及椭圆形和抛物线部分偏微分方程和运算符理论中引起的谐波分析问题。 这些问题很有趣，因为它们对现实世界现象的适用性（见下文），而且还因为它们之间的联系很深。 拟议研究的主要方向包括以下内容：1。要开发和应用``t1/tb''（即卡莱森措施）标准，用于偏椭圆形方程和系统边界问题的溶解度。 3。为了获得傅立叶变换的尖锐衰减，并将这些问题应用于上面提到的晶格问题和法尔康纳的距离问题，以解决谐波分析的问题，并在谐波分析领域及其在与几何学的互动和互动中的相互作用。 （即，映射将一个函数构成另一个功能），将它们分解为较小的组成部分，这些零件易于理解，然后重新组装零件。 thename本身是通过类似于音乐声into的各种频率组件的分解或``谐波''。几次措施理论涉及研究集合的几何特性之间的关系及其``措施''（后者是长度，区域和体积的概念的综合性）。 椭圆形和抛物线类型的PartialDifferentifential方程描述了现实世界中各种各样的现象，包括静电，某些流体流动和弹性变形以及各种扩散程序，例如热量的传导，地下水的流动，​​地下水的流动，​​某些phenomena在人群生物学的数学理论中产生的某些phenomena以及金融市场的数学理论以及金融市场的估计。 在过去的十年中，这些不同的数学子场之间的相互作用被证明是调查的敏捷基础，还有许多令人兴奋的工作要做。