Hua operators on bounded homogeneous domains in \(\mathbb{C}^ n\) and alternative reproducing kernels for holomorphic functions
A canonical system of differential operators for bounded homogeneous domains (more general than the Hermite symmetric domains) in \(\mathbb{C}^n\) is defined by contracting the operator \(\partial\overline\partial\) with the holomorphic curvature tensor, which generalizes the classical Hua operators in the Cartan domains. Functions annihilated by this system are called Hua-harmonic. The main result proved here is the possibility of representing them as ``Poisson integrals'', more precisely, every bounded Hua-harmonic function is a Poisson integral over the Bergman-Shilov boundary of a unique \(L^\infty\) function with respect to the Poisson kernel of the Laplace-Beltrami operator. The authors also consider harmonic functions defined as the null space of a single real differential operator invariant under a particular solvable Lie group which acts transitively on the bounded homogeneous domain. A result in this respect is that there exists an operator satisfying the following three conditions: annihilating holomorphic functions; satisfying the Hörmander condition; having the Bergman-Shilov boundary as its maximal boundary. It follows that the corresponding bounded harmonic functions are in one-to-one correspondence with the \(L^\infty\) functions on the Bergman-Shilov boundary under the Poisson integral.
scientific article; zbMATH DE number 1106590
scientific article