High-power asymptotics of some weighted harmonic Bergman kernels

For weights ρ   which are either radial on the unit ball or depend only on the vertical coordinate on the upper half-space, we describe the asymptotic behaviour of the corresponding weighted harmonic Bergman kernels with respect to ραρα as α→+∞α→+∞. This can be compared to the analogous situation for the holomorphic case, which is of importance in the Berezin quantization as well as in complex geometry.