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S. HofmannUniversity of MissouriDMS 245401 在这个项目中，我将考虑几何分析和几何测度理论以及椭圆和抛物型偏微分方程和算子理论中出现的调和分析问题。 这些问题很有趣，因为它们适用于现实世界的现象（见下文），而且还因为它们之间的深刻联系。 拟议研究的主要方向包括以下内容： 1. 开发和应用椭圆方程和系统的散度边界问题可解性的“T1/Tb”（即卡尔森测度）准则。 2. 处理一致可校正集理论中的各种问题，该理论在椭圆和抛物型偏微分方程中的应用，以及拟共形映射理论中的各种问题。 3. 获得傅里叶变换的尖锐平均衰减估计，并将其应用于具体问题，包括格点问题和福尔科纳距离问题。 如上所述，我建议研究调和分析领域的问题及其在几何测度理论和椭圆和抛物型偏微分方程理论中的应用和相互作用。 粗略地说，在调和分析中，人们研究函数和“算子”的性质 （即，将一个函数转换为另一个函数的映射）通过将它们分解为更小的、更容易理解的组成部分，然后重新组装这些部分。 该名称本身是通过类比将音乐声音分解为其各种频率分量或“谐波”而产生的。几何测度理论涉及对集合的几何性质与其“测度”（后者是长度、面积和体积概念的概括）之间关系的研究。 椭圆型和抛物型偏微分方程描述了现实世界中的各种现象，包括静电、某些流体流动和弹性变形，以及各种扩散过程，例如热传导、地下水流动、人口生物学数学理论中出现的某些现象以及金融市场中的期权定价。 在过去的十年中，这些不同数学子领域之间的相互作用已成为研究的沃土，还有许多令人兴奋的工作有待完成。