High-power asymptotics of some weighted harmonic Bergman kernels

For~weights $\rho$ 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 $\rho^\alpha$ as $\alpha\to+\infty$. 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.
Comment: 15 pages; submitted to J. Funct. Anal. on February 19, 2015