One of the most fundamental theorems in harmonic analysis on Riemannian symmetric spaces is the Helgason theorem which states that any harmonic function may be represented as a Poisson integral over the Furstenberg boundary. Here, "harmonic" means that the function is annihilated by all of the invariant differential operators which have no constant term. For a Hermitian symmetric space, there are generalizations of this theorem which replace the Furstenberg boundary with the Shilov boundary and the invariant differential operators with larger systems such as the Hua-Johnson-Koranyi operators or the Berline-Vergne operators or the Lassalle operators. In this proposal we propose generalizing the Helgason theory, as well as the Hua-Johnson-Koranyi theory, to the context of homogeneous Kaehler manifolds.
Harmonic functions occur repeatedly in science and engineering, in contexts such as heat flow, electricity, wave propagation and particle theory, just to name a few. Many attempts to understand the fundamental properties of nature are based on ever more complicated spaces. The solutions to many important physical and engineering problems on such spaces involve harmonic functions. While the current techniques work reasonably well for Riemannian symmetric spaces, it it is clear that fundamental advances in the current technology must be made if the spaces become even slightly more complicated. Furthermore, the required new technology will certainly yield new information and new techniques in harmonic theory on Riemannian symmetric spaces as well. We propose continuing our previous work which is aimed at the development of this technology.