Mathematical Sciences: Function Theory on Symmetric Spaces

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Koranyi 8901496 This work seeks to exploit the connections between the mathematical theories of differential geometry, representation theory and classsical analysis. It is part of a continuing program of research combining classical topics such as Bessel functions. Hankel transforms and Luzin's area theorem with more recent topics and their generalizations such as quasi-symmetric spaces and more general nilpotent continuous groups. A relatively new thrust in the research concerns the development of a theory of quasiconformal mapping on Cauchy-Riemann manifolds, especially the Heisenberg group. Work of a continuing nature will also be carried out in the study of function theory on symmetric cones. Most of the earlier research on generalized hypergeometric functions in the case of real and complex matrices has been extended to arbitrary symmetric cones. What must be done now is to tie together the various existing theories and find the most general appropriate setting. A new idea is being considered in the still open problem of finding a generalization of Luzin's area integral for Riemannian symmetric spaces. It is based on the construction of a jump process which concentrates in neighborhoods of the distinguished boundary much less rapidly than Brownian motion but for which the harmonic functions are still martingales. There are many applicatons to boundary properties of solutions of differential equations which will not be resolved until successful Fatou-type theorems can be developed in this setting.