Riesz $α$-capacity of Cantor sets and cyclicity in Dirichlet-type spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913024270401536 |
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| author | Vavitsas, Dimitrios Wu, Jujie Zarvalis, Konstantinos |
| author_facet | Vavitsas, Dimitrios Wu, Jujie Zarvalis, Konstantinos |
| contents | We examine the threshold of the cyclicity for functions in Dirichlet-type spaces $\mathcal{D}_α$, $α\in(0,1]$. Given a fixed $α^{*}\in(0,1]$, we construct a holomorphic function $f\in\mathcal{D}_{α^{*}}$ which is cyclic in $\mathcal{D}_α$ for all $α<α^{*}$, but fails to be cyclic in $\mathcal{D}_{α^{*}}$. This function serves as a counterexample to the persistence of cyclicity at the critical index $α^{*}$. Throughout the construction process, we work with generalized Cantor sets and study their Riesz $α$-capacity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_10324 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Riesz $α$-capacity of Cantor sets and cyclicity in Dirichlet-type spaces Vavitsas, Dimitrios Wu, Jujie Zarvalis, Konstantinos Complex Variables Functional Analysis Primary: 30C85, 46E20, 47A16, Secondary: 30H99, 31A15 We examine the threshold of the cyclicity for functions in Dirichlet-type spaces $\mathcal{D}_α$, $α\in(0,1]$. Given a fixed $α^{*}\in(0,1]$, we construct a holomorphic function $f\in\mathcal{D}_{α^{*}}$ which is cyclic in $\mathcal{D}_α$ for all $α<α^{*}$, but fails to be cyclic in $\mathcal{D}_{α^{*}}$. This function serves as a counterexample to the persistence of cyclicity at the critical index $α^{*}$. Throughout the construction process, we work with generalized Cantor sets and study their Riesz $α$-capacity. |
| title | Riesz $α$-capacity of Cantor sets and cyclicity in Dirichlet-type spaces |
| topic | Complex Variables Functional Analysis Primary: 30C85, 46E20, 47A16, Secondary: 30H99, 31A15 |
| url | https://arxiv.org/abs/2604.10324 |