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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/2412.13854 |
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Table of Contents:
- In this article, we investigate the connection between certain real variable things and the Bergman theory. We first use Hardy-type inequalities to give an $L^2$ Hartogs-type extension theorem and an $L^p$ integrability theorem for the Bergman kernel $K_Ω(\cdot,w)$. We then use the Sobolev-Morrey inequality to show the absolute continuity of Bergman kernels on planar domains with respect to logarithmic capacities. Finally, we give lower bounds of the minimum $κ(Ω)$ of the Bergman kernel $K_Ω(z)$ in terms of the interior capacity radius for planar domains and the volume density for bounded pseudoconvex domains in $\mathbb C^n$. As a consequence, we show that $κ(Ω)\ge c_0 λ_1(Ω)$ holds on planar domains, where $c_0$ is a numerical constant and $λ_1(Ω)$ is the first Dirichlet eigenvalue of $-Δ$.