Asymptotics of reproducing kernels on a plane domain

Abstract:

Let $\Omega$. be a plane domain of hyperbolic type, $|dz|/w(z)$ the Poincaré metric on $\Omega$, and ${K_{\Omega ,q}}(x,\bar y)$ the reproducing kernel for the Hilbert space $\mathcal {A}_q^2(\Omega )$ of all holomorphic functions on $\Omega$ square-integrable with respect to the measure $w{(z)^{2q - 2}}|dz \wedge d\bar z|$. It is proved that \[ \lim \limits _{q \to + \infty } \frac {{{K_{\Omega ,q}}(z,\bar z)w{{(z)}^{2q}}}}{{2q}} = \frac {1}{\pi }.\]