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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.00988 |
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| _version_ | 1866914854200147968 |
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| author | Cho, Hong Rae Park, Soohyun |
| author_facet | Cho, Hong Rae Park, Soohyun |
| 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00988 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pointwise estimates of the Bergman kernel with an exponential weight on the unit ball Cho, Hong Rae Park, Soohyun Complex Variables 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}. |
| title | Pointwise estimates of the Bergman kernel with an exponential weight on the unit ball |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2407.00988 |