Geometric conditions for bounded point evaluations in several complex variables
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912677864931328 |
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| author | Deterding, Stephen |
| author_facet | Deterding, Stephen |
| contents | Let $U$ be a bounded domain in $\mathbb C^d$ and let $L^p_a(U)$, $1 \leq p < \infty$, denote the space of functions that are analytic on $\overline{U}$ and bounded in the $L^p$ norm on $U$. A point $x \in \overline{U}$ is said to be a bounded point evaluation for $L^p_a(U)$ if the linear functional $f \to f(x)$ is bounded in $L^p_a(U)$. In this paper, we provide a purely geometric condition given in terms of the Sobolev $q$-capacity for a point to be a bounded point evaluation for $L^p_a(U)$. This extends results known only for the single variable case to several complex variables. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_23688 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric conditions for bounded point evaluations in several complex variables Deterding, Stephen Complex Variables 32A37 (Primary) 32E30 (Secondary) Let $U$ be a bounded domain in $\mathbb C^d$ and let $L^p_a(U)$, $1 \leq p < \infty$, denote the space of functions that are analytic on $\overline{U}$ and bounded in the $L^p$ norm on $U$. A point $x \in \overline{U}$ is said to be a bounded point evaluation for $L^p_a(U)$ if the linear functional $f \to f(x)$ is bounded in $L^p_a(U)$. In this paper, we provide a purely geometric condition given in terms of the Sobolev $q$-capacity for a point to be a bounded point evaluation for $L^p_a(U)$. This extends results known only for the single variable case to several complex variables. |
| title | Geometric conditions for bounded point evaluations in several complex variables |
| topic | Complex Variables 32A37 (Primary) 32E30 (Secondary) |
| url | https://arxiv.org/abs/2507.23688 |