Harmonic Analysis and the Theory of Elliptic Measure

谐波分析与椭圆测度理论

• 批准号: 0901139
• 负责人: Jill Pipher
• 金额: $ 43.53万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901139&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Theory; Elliptic; Measure

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).There are two parts to the research program in this proposal. Part one concerns multi-parameter Fourier analysis, the higher dimensional (or product) theory with origins in the theory of several complex variables, convergence of multiple Fourier series and analysis on symmetric spaces. Some of the modern questions proposed here are related to singular integrals, Hardy spaces, averaging, bilinear operators, and reach into probability theory through multi-parameter martingales, and higher dimensional Brownian sheets. Part two of this proposal concerns the theory of second order elliptic partial differential equations, in both divergence and non-divergence form. The questions are mainly centered on the fine properties of the elliptic measure associated with the solvability of boundary value problems and are posed in two settings: when the operator itself has non-smooth coefficients, or for smooth operators in domains whose boundary is less regular yet satisfies some geometric conditions (Lipschitz, chord-arc, Reifenberg flat, for example).In 1811, Joseph Fourier proposed a mathematical theory of heat which took the point of view that heat was a ``flow" and should be described by differential equations. This was the starting point of the theory now called Fourier analysis, a field which has an impact on every area of mathematics and the physical sciences. In this proposal, we consider some basic open problems related to such differential equations, and try to understand how their solutions depend on boundary conditions. Solving these problems may lead to a better understanding of how to obtain approximations to solutions (using computational tools) when it is impossible to write down a formula which describes them exactly. This grant will partially support graduate students and undergraduates in this field by providing them the opportunity to do research in this area (individually and in teams) and to travel to conferences where they can present their own results.

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。该提案中的研究计划分为两部分。第一部分涉及多参数傅里叶分析，起源于多复变量理论的高维(或乘积)理论，多重傅立叶级数的收敛和对称空间上的分析。这里提出的一些现代问题涉及奇异积分、Hardy空间、平均、双线性算子，以及通过多参数鞅和高维布朗单达到概率论。本建议的第二部分涉及二阶椭圆型偏微分方程的理论，既有散度形式的，也有非散度形式的。这些问题主要集中在与边值问题的可解性有关的椭圆测度的精细性质上，并在两种情况下提出：当算子本身具有非光滑系数时，或对于边界较不规则但满足某些几何条件(例如，Lipschitz、弦弧、Reifenberg Flat)的区域中的光滑算子。1811年，约瑟夫·傅立叶提出了热的数学理论，该理论认为热是一种‘流’，应该用微分方程来描述。这是现在被称为傅里叶分析的理论的起点，这个领域对数学和物理科学的每个领域都有影响。在这个方案中，我们考虑一些与这类微分方程有关的基本公开问题，并试图理解它们的解是如何依赖于边界条件的。解决这些问题可能导致更好地理解当不可能写下准确描述它们的公式时如何(使用计算工具)获得解的近似值。这笔赠款将部分支持这一领域的研究生和本科生，为他们提供在这一领域(个人和团队)进行研究的机会，并参加会议，在那里他们可以展示自己的成果。