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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.00988 |
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Table of Contents:
- We consider the weighted Bergman space $A^2_ψ(\Bn)$ of all holomorphic functions on $\Bn$ square integrable with respect to a particular exponential weight measure $e^{-ψ} dV$ on $\Bn$, where \begin{align*} ψ(z):=\frac{1}{1-|z|^2}. \end{align*} We prove the following estimate for the Bergman kernel $K_ψ(z,w)$ of $A^2_ψ(\Bn)$: \begin{align*} |K_ψ(z,w)|^2\le C\frac{e^{ψ(z)+ψ(w)}}{{\rm Vol}(B_ψ(z,1)){\rm Vol}(B_ψ(w, 1))}e^{-\varepsilon d_ψ(z,w)}, \quad z, w\in\Bn, \end{align*} where $d_ψ$ is the Riemannian distance induced by the potential function $ψ$ and $B_ψ(z,1)$ is the $d_ψ$-ball of center $z$ and radius $1$. The result is motivated by Christ \cite{Chr}.