Mathematical Sciences: Function Theory on Symmetric Spaces.

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This project represents a continuation of work on the theory of quasiconformal mappings on the Heisenberg group. Problems of global distortion, extension theorems and relaxation of smoothness conditions present in existing results will be treated. The same problems will also be considered on general CR manifolds with positive Levi form with applications to complex variable theory. A second thrust will be made in the direction of harmonic analysis on symmetric cones. With the use of Jordan algebras a theory of special functions will be developed. In addition, boundary behavior of harmonic functions on symmetric spaces will be analyzed through the use of Tits buildings. In this context a generalized Luzin area function will be sought. Finally, the generalization of the geometric properties of the Heisenberg group to so-called H-type groups will be carried out. The work planned relates to several areas of mathematics, including classical function theory and quasiconformal mapping. Other applications to the gamma function, boundary behavior in potential theory and the Plancherel theorem can be made. New insights into the holomorphic equivalence problem are possible