Singular Integral Operators in Several Complex Variables

多复变函数论中最重要的偏微分算子是柯西-黎曼(CR)算子和切向CR-算子，它们是通过将CR-算子限制在n维复空间中的一个曲面上而得到的。这个项目的目的是以一种重要的方式推广关于切向CR-算子和相关偏微分方程组的已知结果。一般情况下，这个系统不存在唯一解，所以人们所能做的最好的事情就是找到与算子的零空间正交的唯一平方可积解。因此，一个基本的研究对象是Szego投影算子，它是平方可积函数空间在切CR算子的零空间上的正交投影。对于二维复空间中有限类型的超曲面(这意味着它们不是太平的)，这个运算符被很好地理解。这个项目的一部分研究了两维空间中无限类型曲面的Szego投影算子。这种方法是根据对内核的积分来考虑运算符，然后估计内核函数。在有限类型情况下使用的方法不能扩展到这种情况，因此需要新的想法。主要研究人员还打算研究高维凸曲面上的Szego投影算子的核。在这种情况下，算子被很好地理解，尽管以前的方法使用了Bergman核的知识，而不是直接分析积分核。对后者的研究应该会对在这种情况下产生的非各向同性度量有更多的洞察。最后，我们将研究复二维空间中模型非(伪)凸曲面的CR算子本身，其中Szego核的估计存在，但对CR算子本身知之甚少。偏微分方程组为描述变量之间的关系提供了强大的语言。一些人模拟热流，另一些人关注波的传播，还有一些人本身就是研究的对象。对于任何偏微分，人们都感兴趣的是根据给定的初始数据来求解未知函数。其目的是确定解存在的条件、解唯一的条件以及给定数据的属性与解的属性之间的关系。虽然这个项目研究的是对纯数学而不是应用数学有根本兴趣的偏微分方程式，但开发的新技术将会更广泛地适用。

The most important partial differential operators in the theory of functions of several complex variables are the Cauchy-Riemann (CR) operator and the tangential CR- operator, obtained by restricting the CR-operator to a surface in complex n-dimensional space. This project aims to extend in a significant way known results concerning the tangential CR-operator and the associated system of partial differential equations. In general, this system does not have a unique solution, so the best that one can do is to find the unique square-integrable solution orthogonal to the null-space of the operator. For this reason a fundamental object of study is the Szego projection operator, which is the orthogonal projection of the space of square-integrable functions onto the null space of the tangential CR-operator. This operator is well understood in the case of hypersurfaces in two-dimensional complex space that are of finite type (which means that they are not "too flat"). Part of this project examines the Szego projection operator for infinite-type surfaces in two-space. The approach is to think of the operator in terms of integration against a kernel and then to estimate the kernel function. The methods used in the finite-type case do not extend to this setting, so new ideas are needed. The principal investigator also intends to study the kernel for the Szego projection operator for convex surfaces in higher dimensions. The operator is reasonably well understood in this situation, though previous approaches have used knowledge of the Bergman kernel rather than direct analysis of the integral kernel. An examination of the latter should give additional insight into the nonisotropic metric that arises in this case. Finally, the CR- operator itself will be studied for a model non-(pseudo)convex surface in complex two-space for which estimates on the Szego kernel exist, but for which little is known about the CR-operator itself.Partial differential equations provide a powerful language for describing relationships between changing quantities. Some model heat flow, others are concerned with wave propagation, and yet others are just objects of study in their own right. For any partial differential one is interested in solving for unknown functions in terms of given initial data. The objective is to determine conditions under which a solution exists, conditions under which a solution is unique, and relationships between the properties of the given data and properties of the solution. Although this project studies partial differential equations that are of fundamental interest in pure rather than applied mathematics, the new techniques developed will be much more broadly applicable.