Heat Equations, Boundary Operators and CR Geometry in Complex Analysis

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。主要研究多变量加权Cauchy-Riemann方程的正则性和解的衰减估计，以及伪凸域边界上的切向Cauchy-Riemann方程。当区域无界时，他将通过其热方程研究Kohn Laplacian，并发展必要的调和分析和正则性理论，以获得对热核及其导数的点估计。在cr流形紧的情况下，问题将取决于cr流形是否是伪凸域的边界，或者至少有两个完全实的切束方向。在前一种情况下，检验将集中在复Green算子的紧性关系、Stein邻域基的存在性以及被称为“P”子“q”性质的某种性质上。在后一种情况下，首席研究员将对切向柯西-黎曼算子的闭合范围性质进行分析。几个复杂变量的一个基本问题是解切向柯西-黎曼方程。也许这个问题最具挑战性的方面是缺乏椭圆性和几何对分析的影响。然而，对切向Cauchy-Riemann方程的研究导致了对偏微分方程理解的重大进展，也许最引人注目的例子是局部不可解线性偏微分方程的存在（即Lewy的例子）和伪微分算子的发展。这个项目将有助于理解cr流形几何对切向柯西-黎曼算子正则性理论的影响。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The principal investigator will undertake a study of regularity properties and decay estimates of solutions to the weighted Cauchy-Riemann equations in several variables as well as the tangential Cauchy-Riemann equations on the boundaries of pseudoconvex domains. When the domain is unbounded, he will investigate the Kohn Laplacian via its heat equation and develop the necessary harmonic analysis and regularity theory to obtain pointwise estimates on the heat kernel and its derivatives. In the case that the CR-manifold is compact, the questions will depend on whether the CR-manifold is the boundary of a pseudoconvex domain or has at least two totally real directions to the tangent bundle. In the former case, the examination will focus on the relationship of compactness of the complex Green operator, the existence of Stein neighborhood bases, and a certain property known as the "P" sub "q" property. In the latter case, the principal investigator will undertake an analysis of the closed range properties of the tangential Cauchy-Riemann operator. A fundamental problem in several complex variables is to solve the tangential Cauchy-Riemann equations. Perhaps the most challenging aspects to this problem are the lack of ellipticity and the influence of the geometry on the analysis. The study of the tangential Cauchy-Riemann equations, however, has led to significant advances in the understanding of partial differential equations, perhaps the most striking examples being the existence of a locally nonsolvable linear partial differential equation (i.e., Lewy's example) and the development of pseudodifferential operators. This project will contribute to the understanding of the effect of the geometry of a CR-manifold on the regularity theory of the tangential Cauchy-Riemann operators.