In many senses the harmonic analysis of Euclidean space is well developed. The theories of fractional integration, singular integrals, parametrices for constant coefficient partial differential operators, and related matters are well understood. In all of these theories, translation invariance plays a crucial role. In particular, Fourier analysis is a central tool in all of this work. Many physical problems - the flow of gas around an obstacle, turbulence, problems of flux, Dirichlet and Neumann problems, etc. - have as their natural setting a smoothly bounded domain (where the necessary degree of smoothness may vary with the application). The canonical setting for many of the basic problems of harmonic analysis and of complex function theory is also a domain in space. Thus, it becomes appropriate to develop analytic tools tailored to the geometry of domains. This leads to reconsidering classical function spaces on these domains and to studying them with the use of invariant metrics. This project will support an NSF-CBMS Regional Research Conference in the Mathematical Sciences on New Function Spaces and Geometric Analysis in Several Complex Variables to be held May 26-31, 1992 at George Mason University. Professor Steven Krantz of Washington University will be the principal lecturer. To stimulate interest and activity in mathematical research, the National Science Foundation each year supports a number of NSF- CBMS Regional Research Conferences in the Mathematical Sciences. Each five-day conference features a distinguished lecturer who delivers ten lectures on a topic of important current research in one sharply focused area of the mathematical sciences. The lecturer subsequently prepares an expository monograph based upon these lectures, which is normally published by the American Mathematical Society or the Society for Industrial and Applied Mathematics, or jointly by the American Statistical Association and the Institute of Mathematical Statistics. Certain features differentiate these conferences from typical research conferences. These are: (1) Focus on a single important and timely area of research by a leading practitioner, (2) Continued effect and local stimulation through regional emphasis, (3) Panel review for quality, breadth, and timeliness, and (4) Published monographs for a wider audience.
