High-power asymptotics of some weighted harmonic Bergman kernels

TL;DR

This paper investigates the asymptotic behavior of weighted harmonic Bergman kernels with specific radial or vertical weights as the weight parameter grows large, drawing parallels to the holomorphic case relevant in geometry and quantization.

Contribution

It provides a detailed description of the asymptotics of these kernels for certain weights, extending understanding from the holomorphic setting to harmonic functions.

Findings

Asymptotic formulas for weighted harmonic Bergman kernels

Comparison with holomorphic Bergman kernel asymptotics

Implications for Berezin quantization and complex geometry

Abstract

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.