Hua Operators on Bounded Homogeneous Domains in Cnand Alternative Reproducing Kernels for Holomorphic Functions

Let D be a bounded homogeneous domain in Cn. (Note that D is not assumed to be Hermitian-symmetric.) In this work we are interested in studying various classes of “harmonic” functions on D and the possibility of representing them as “Poisson integrals” over the Bergman-Shilov boundary. One such class of harmonic functions is the “Hua-harmonic” functions. Specifically, by forming a contraction of ∂∂ with the holomorphic curvature tensor, we define a canonical system of differential operators which generalizes the classical Hua system. This system is invariant under all bi-holomorphisms 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 uniqueL∞function against the Poisson kernel for the Laplace-Beltrami operator. We also consider spaces of harmonic functions defined as the kernel of a single real differential operator which is invariant under a particular solvable Lie group which acts transitively on D. We show that there exists such an operator which (a) annihilates holomorphic functions, (b) satisfies the Hormander condition, and (c) has the Bergman-Shilov boundary as its maximal boundary. It follows that the corresponding bounded harmonic functions are in one-to-one correspondence with theL∞functions on the Bergman-Shilov boundary under Poisson integration.