Work done on this project combines elements of the theory of several complex variables with those of harmonic analysis on Euclidean domains with applications to partial differential equations. During the past five years or so, developments in the analysis of functions of two complex variables on pseudoconvex domains has reached the level of completeness comparable to what is known for strictly pseudoconvex domains in arbitrary dimension. Regularity results for the Bergman and Szego projections, sharp estimates for the basic differential operators and characterizations of zero varieties are well understood. The present work will continue the task of extending these kinds of results to domains (of finite type) of dimension greater than two. A second fundamental problem concerns metrics on the boundary of domains. If the domain has finite type, then the metrics, although not unique, are classified. For domains of general type, it is not even known if there is a reasonable boundary metric which can be used to analyze some of the important analytic objects on the domain. Work is proceeding through the study of specific domains to clarify the right questions to be studied. Work on Fourier analysis is related to Fourier transforms of measures defined on submanifolds in Euclidean space. Estimates of the transform have important applications in a variety of problems ranging from number theory to partial differential equations. The best results to date assume that the manifold is the boundary of a convex domain. Little is known if the domain is nonconvex or if the manifold has high codimension. Although some information is available concerning the transform at infinity, much remains to be done in obtaining geometric description and interpretation of the transform and its antecedent measure.
