Mathematical Sciences: Poisson Integrals and Cayley Transforms on Bounded Homogeneous Domains in Cn

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Penney will continue his investigation to determine what aspects of the harmonic analysis of Hermitian symmetric spaces generalize to non-symmetric situations. Particular questions yet to be resolved are (a) how to characterize the Poisson integrals of hyperfunctions on the boundary and (b) how to treat unbounded domains. The first step will be to determine if there exist on a bounded homogeneous domain a system of differential operators which annihilates the Gindikin-Poisson kernel. The basic idea behind Penney's project can be seen already in the motion group of the line, which consists of translations and reflection about zero. The quotient of the motion group by the reflection subgroup is naturally isomorphic to the line. Similarly, more general geometric objects can be realized as the quotient of a transformation group by a certain subgroup. This introduces natural classes of group invariant operators on the geometric space, and analysis is developed using eigenfunctions of the group invariant operators.