Some Fourier Analysis Problems Related to Several Complex Variables

与多个复变量相关的一些傅立叶分析问题

• 批准号: 9000968
• 负责人: Der-Chen Chang
• 金额: $ 3.81万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-04-15 至 1992-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9000968&HistoricalAwards=false
• 关键词: Some; Fourier; Analysis; Problems; Related

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.

虽然这项数学研究涉及的理论， 多复空间中的光滑有界域 变量，重点是研究解决方案的行为 一阶和二阶偏微分方程 主要 主题是通过投影确定函数类 函数到微分方程的解空间上 方程 具体地说，人们想知道函数是否在 经典的勒贝格类，或那些保持器 连续投影到同一类型的函数上。 答案 取决于边界的平滑度，可能还取决于 环境空间的维度。 在强伪- 凸域，结果是相当完整的。 对于弱伪- 凸域有限型在两个复杂的维度，相同的 可以说。 这是目前的分界线。 这个项目首先要做的工作将占用 Lebesgue空间的正则性问题 一个. 具体做法将侧重于 域是Siegel上半平面的特定情况。 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。 第二条调查路线将集中在对 两个复形中Szego核（投影）的正则性 具有无限型边界点的域的维数。 这些类型的边界具有对扩张的抵抗力， 这是唯一能证明正则性的方法 一些 这里已经取得了进展，表明前几个术语 Szego内核的大小是可以计算的。