Multi-parameter singular integrals

研究者将研究Calderón-Zygmund奇异积分理论的多参数类似物，它显著地推广了著名的奇异积分积理论。一个关键的起点将是多参数carnot - carathacimodory（或亚黎曼）几何（由向量场定义的几何）的情况。在carnot - carathimodory几何中，关于Calderón-Zygmund奇异积分的类比已经有了一个合理的猜想：这个猜想推广了一些已知的有用的奇异积分类型。Calderón-Zygmund奇异积分理论在广泛的数学领域中得到了广泛的应用。然而，当底层几何是多参数时，没有已知的Calderón-Zygmund理论的类比（除了产品类型的情况）。最近的研究表明，当底层几何由多参数carnot - carathacimodory几何给出时，模拟可能是可行的。谐波分析，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得到了广泛的应用。谐波分析的各种方法为许多未来的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。更一般地说，它适用于由向量场定义的偏微分方程：这是一个在数学、金融和流体动力学中有影响的理论。目前将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特定应用的“多参数”理论。这个项目的主要目的是发展这样一个理论，它将在各种各样的情况下有用——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积极研究和培训谐波分析，特别是谐波分析与偏微分方程的应用。这包括许多与研究生和访问博士后学者的积极讨论和合作。

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.