Some Fourier Analysis Problems Related to Several Complex Variables

点击翻译按钮获取中文摘要

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.