Some Fourier Analysis Problems Related to Several Complex Variables

虽然这一数学研究关注的是复数变量空间中光滑有界域的理论，但重点是研究一阶和二阶偏微分方程解的行为。主要的主题是通过将函数投影到微分方程解的空间来确定函数的类别。具体来说，人们想知道经典勒贝格类中的函数，或者那些Holder连续类中的函数是否投影到同一类型的函数上。答案取决于边界的平滑度，也可能取决于周围空间的维度。在强伪凸域的情况下，结果是相当完整的。对于两个复维的有限型弱伪凸域，也可以这样说。这是目前的分界线。在这个项目中首先要做的工作是研究小于1次幂的勒贝格空间的正则性问题。一种具体的方法是关注域是西格尔上半平面的具体情况。在这种情况下，谐波分析的强大机制可以利用现有的哈代空间理论。第二方面的研究将集中在具有无限型边界点的域的二维Szego核（投影）的正则性的研究上。这些类型的边界具有抵抗扩张的能力——这是目前唯一可用来证明规律性的技术。这里已经取得了一些进展，表明szgo内核的前几个项实际上是可以计算的。

Although this mathematical research concerns theory on smoothly bounded domains in the space of several complex variables, the focus is on studying the behavior of solutions of first- and second-order partial differential equations. The main theme is to determine function classes determined by projecting functions onto the spaces of solutions of the differential equations. Specifically, one would like to know if functions in the classical Lebesgue classes, or those which are Holder continuous project onto functions of the same type. The answer depends on the smoothness of the boundary and, possibly, on the dimension of the ambient space. In the case of strongly pseudo- convex domains, results are quite complete. For weakly pseudo- convex domains of finite type in two complex dimensions, the same can be said. This is the current dividing line. The work to be done in this project first will take up the regularity question for the Lebesgue spaces for powers less than one. A concrete approach will be made by focusing on the specific case where the domain is the Siegel upper half plane. In this setting the powerful machinery of harmonic analysis can be brought to bear using existing theory of Hardy spaces. A second line of investigation will center on studies of the regularity of the Szego kernel (projection) in two complex dimensions for domains with a boundary point of infinite type. These types of boundaries have a resistance to dilation - at present the only technique available to prove regularity. Some progress has been made here in showing that the first few terms of the Szego kernel can actually be computed.