Multi-parameter singular integrals

研究人员将研究奇异积分的Calderón-Zygmund理论的多参数类似物，该理论显着推广了奇异积分的着名乘积理论。 一个关键的起点将是多参数卡诺-卡拉西奥多里（或亚黎曼）几何（由向量场定义的几何）的情况。 在卡诺-卡拉西奥多里几何中，已经有一个关于类似卡尔德龙-齐格蒙德奇异积分的合理猜想：这个猜想推广了一些已知的有用的奇异积分类型。 卡尔德龙-齐格蒙德奇异积分理论在广泛的数学中有着广泛的应用。 然而，当基础几何是多参数时，没有已知的卡尔德龙-齐格蒙德理论的类似物（在乘积型情况之外）。 最近的工作表明，当多参数卡诺-卡拉西奥多里几何给出了基础几何时，类似的东西可能就可以实现了。调和分析，更具体地说，奇异积分理论，在数学、物理、金融和生物学的其他领域有着广泛的应用。 谐波分析中的各种方法提供了许多未来科学应用的前景。 特别是，这个项目的研究与数学、金融和数学物理的其他领域有几个联系。 最直接的是，它可以应用于多复变量理论。 更一般地说，它适用于由向量场定义的偏微分方程：一个在数学金融和流体动力学中有影响的理论。 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于具体应用的“多参数”理论。 这个项目的主要目的是发展这样一种理论，它将在各种各样的情况下是有用的-可能解决一些悬而未决的问题。 该项目将帮助威斯康星-麦迪逊大学继续积极开展调和分析-特别是调和分析在偏微分方程中的应用-方面的研究和培训工作。 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。

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.