Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.13854 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915069762207744 |
|---|---|
| author | Chen, Bo-Yong Xiong, Yuanpu |
| author_facet | Chen, Bo-Yong Xiong, Yuanpu |
| 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 $-Δ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Real Variable Things in Bergman Theory Chen, Bo-Yong Xiong, Yuanpu Complex Variables 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 $-Δ$. |
| title | Real Variable Things in Bergman Theory |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2412.13854 |