Advanced Search

We consider holomorphic functions and complex harmonic functions on some balls, including the complex Euclidean ball, the Lie ball and the dual Lie ball. After reviewing some results on Bergman kernels and harmonic Bergman kernels for these balls, we consider harmonic continuation of complex harmonic functions on these balls by using harmonic Bergman kernels. We also study Szegő kernels and harmonic Szegő kernels for these balls.

Let L(z) be the Lie norm on $\tilde{}={ℂ}^{n+1}$ and L*(z) the dual Lie norm. We denote by ${}_{\Delta }\left(\tilde{B}\left(R\right)\right)$ the space of complex harmonic functions on the open Lie ball $\tilde{B}\left(R\right)$ and by $Ex{p}_{\Delta }(\tilde{};(A,L*))$ the space of entire harmonic functions of exponential type (A,L*). A continuous linear functional on these spaces will be called a harmonic functional or an entire harmonic functional. We shall study the conical Fourier-Borel transformations on the spaces of harmonic functionals or entire harmonic functionals.