HUA OPERATORS ON BOUNDED HOMOGENEOUS-DOMAINS IN C-N AND ALTERNATIVE REPRODUCING KERNELS FOR HOLOMORPHIC-FUNCTIONS

Let D be a bounded homogeneous domain in C-n. (Note that D is not assu med to be Hermitian-symmetric.) In this work we are interested in stud ying various classes of ''harmonic'' functions on D and the possibilit y of representing them as ''Poisson integrals'' over the Bergman-Shilo v boundary. One such class of harmonic functions is the ''Hua-harmonic '' functions. Specifically, by forming a contraction of partial deriva tive partial derivative with the holomorphic curvature tensor, we defi ne a canonical system of differential operators which generalizes the classical Hua system. This system is invariant under all bi-holomorphi sms of D. The Hua-harmonic functions are, by definition, the nullspace of this system. Our main result concerning this system is that every bounded Hua-harmonic Function is the Poisson-integral over the Bergman -Shilov boundary of a unique L-infinity function against the Poisson k ernel for the Laplace-Beltrami operator. We also consider spaces of ha rmonic functions defined as the kernel of a single real differential o perator which is invariant under a particular solvable Lie group which acts transitively on D. We show that there exists such an operator wh ich (a) annihilates holomorphic functions, (b) satisfies the Hormander condition, and (c) has the Bergman-Shilov boundary as its maximal bou ndary. It follows that the corresponding bounded harmonic functions ar e in one-to-one correspondence with the L-infinity Functions on the Be rgman-Shilov boundary under Poisson integration. (C) 1997 Academic Pre ss.