Multi-parameter singular integrals

研究人员将研究Calderón-Zygmund奇异积分理论的多参数类比，它显著推广了著名的奇异积分乘积理论。一个关键的起点将是多参数Carnot-Carathéodory(或次Riemannian)几何(由矢量场定义的几何)的情况。关于Calderón-Zygmund奇异积分在Carnot-Carathéodory几何中的类比，已经有一个合理的猜想：一个推广了许多已知和有用的奇异积分类型的猜想。奇异积分的Calderón-Zygmund理论在广泛的数学中得到了广泛的应用。然而，当基本几何是多参数时，没有已知的类似Calderón-Zygmund理论(产品类型的情况之外)。最近的工作表明，当基础几何由多参数Carnot-Carathéodory几何给出时，可能会达到类似的结果。调和分析，更具体地说，奇异积分理论，在数学、物理、金融和生物的其他领域得到了广泛的应用。谐波分析中的各种方法为未来的许多科学应用提供了希望。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，它可以应用于几个复变量的理论。更广泛地说，它适用于由矢量场定义的偏微分方程式：这是一种在数学金融和流体动力学中有意义的理论。目前，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个理论，它将在广泛的各种情况下有用--可能解决一些悬而未决的问题。该项目将有助于继续在威斯康星大学麦迪逊分校进行积极的调和分析研究和培训小组--特别是应用于偏微分方程式的调和分析。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.