An important property of the reproducing kernel of D2(Ω,ρ) is obtained and the reproducing kernels for D2(Ω,ρ) are calculated when Ω = Bn × Bn and ρ are some special functions.A reproducing kernel is used to construct a semi-positive definite matrix and a distance function defined on Ω×Ω.An inequality is obtained about the distance function and the pseudo-distance induced by the matrix.
We obtain the explicit formulae for the harmonic Bergman kernels of Bn\{0} and Rn\Bn and study the connection between harmonic Bergman kernel and weighted harmonic Bergman kernel.We also get the explicit formula for the weighted harmonic Bergman kernel of Bn\{0} with the weight 1/|x|4.
