Multi-parameter singular integrals

多参数奇异积分

• 批准号: 1066020
• 负责人: Brian Street
• 金额: $ 10万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1066020&HistoricalAwards=false
• 关键词: Multi; parameter; singular; integrals

The investigator will study multi-parameter analogs of the Calderón-Zygmund theory of singular integrals, which significantly generalize the well-known product theory of singular integrals. A critical starting point will be the case of multi-parameter Carnot-Carathéodory (or sub-Riemannian) geometry (a geometry defined by vector fields). There is already a reasonable conjecture as to the analog of a Calderón-Zygmund singular integral in the context of Carnot-Carathéodory geometry: a conjecture which generalizes a number of known and useful types of singular integrals. The Calderón-Zygmund theory of singular integrals has found numerous applications in a wide range of mathematics. However, when the underlying geometry is multi-parameter, there is no known analog of the Calderón-Zygmund theory (outside of the product-type situation). Recent work shows that an analog might be in reach when the underlying geometry is given by a multi-parameter Carnot-Carathéodory geometry.Harmonic analysis, and more specifically the theory of singular integrals, has found a wide variety of applications in other areas of mathematics, physics, finance, and biology. The diverse methods in harmonic analysis offer the promise of many future applications in the sciences. The research in this project, in particular, has several connections to other areas of mathematics, finance, and mathematical physics. Most directly, it has applications to the theory of several complex variables. More generally, it applies to partial differential equations defined by vector fields: a theory which has implications in mathematical finance and fluid dynamics. One of the main current obstacles in the application of the theory of singular integrals to various questions is that there is no suitable "multi-parameter" theory adapted to the particular application. The main purpose of this project is to develop such a theory, which would be useful in a wide variety of situations--potentially addressing a number of open questions. The project will help continue an active research and training group in harmonic analysis--especially harmonic analysis with applications to partial differential equations--at the University of Wisconsin-Madison. This includes many active discussions and collaborations with graduate students and visiting postdoctoral scholars.

研究人员将研究奇异积分 Calderón-Zygmund 理论的多参数类似物，该理论显着概括了众所周知的奇异积分乘积理论。 一个关键的起点是多参数卡诺-卡拉西奥多里（或亚黎曼）几何（由矢量场定义的几何）的情况。 关于卡诺-卡拉西奥多里几何背景下卡尔德隆-齐格蒙德奇异积分的类比，已经存在一个合理的猜想：该猜想概括了许多已知且有用的奇异积分类型。 奇异积分的卡尔德龙-齐格蒙德理论在广泛的数学领域中得到了广泛的应用。 然而，当基础几何形状是多参数时，没有已知的 Calderón-Zygmund 理论的类似物（在产品类型情况之外）。 最近的工作表明，当底层几何由多参数卡诺-卡拉西奥多里几何给出时，可能可以进行类似的分析。调和分析，更具体地说是奇异积分理论，已经在数学、物理、金融和生物学的其他领域找到了广泛的应用。 调和分析中的多种方法为未来许多科学应用提供了前景。 特别是，该项目的研究与数学、金融和数学物理的其他领域有多种联系。 最直接的是，它应用于多个复变量的理论。 更一般地说，它适用于由矢量场定义的偏微分方程：一种对数学金融和流体动力学有影响的理论。 当前奇异积分理论应用于各种问题的主要障碍之一是没有适合特定应用的合适的“多参数”理论。 该项目的主要目的是发展这样一种理论，该理论在各种情况下都很有用——有可能解决许多悬而未决的问题。 该项目将帮助威斯康星大学麦迪逊分校继续活跃的调和分析研究和培训小组，特别是调和分析及其在偏微分方程中的应用。 这包括与研究生和访问博士后学者的许多积极讨论和合作。