Characterizations of Lyapunov domains in terms of Riesz transforms and the Plemelj-Privalov theorem
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| Format: | Preprint |
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2026
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| author | Marín, Juan José Martell, José María Mitrea, Dorina Mitrea, Marius |
| author_facet | Marín, Juan José Martell, José María Mitrea, Dorina Mitrea, Marius |
| contents | We prove several characterizations of $\mathscr{C}^{1,ω}$-domains (aka Lyapunov domains), where $ω$ is a growth function satisfying natural assumptions. For example, given an Ahlfors regular domain $Ω\subseteq{\mathbb{R}}^n$, we show that the modulus of continuity of the geometric measure theoretic outward unit normal $ν$ to $Ω$ is dominated by (a multiple of) $ω$ if and only if the action of each Riesz transform $R_j$ associated with $\partialΩ$ on the constant function $1$ has a modulus of continuity dominated by (a multiple of) $ω$. The proof of this result requires that we establish a higher-dimensional generalization of the classical Plemelj-Privalov theorem, identifying a large class of singular integral operators that are bounded on generalized Hölder spaces. This class includes the Cauchy-Clifford operator and the harmonic double layer operator, among others. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_19143 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterizations of Lyapunov domains in terms of Riesz transforms and the Plemelj-Privalov theorem Marín, Juan José Martell, José María Mitrea, Dorina Mitrea, Marius Classical Analysis and ODEs 26A16, 28A75, 31B10, 42B20, 42B35, 42B37, 15A66 We prove several characterizations of $\mathscr{C}^{1,ω}$-domains (aka Lyapunov domains), where $ω$ is a growth function satisfying natural assumptions. For example, given an Ahlfors regular domain $Ω\subseteq{\mathbb{R}}^n$, we show that the modulus of continuity of the geometric measure theoretic outward unit normal $ν$ to $Ω$ is dominated by (a multiple of) $ω$ if and only if the action of each Riesz transform $R_j$ associated with $\partialΩ$ on the constant function $1$ has a modulus of continuity dominated by (a multiple of) $ω$. The proof of this result requires that we establish a higher-dimensional generalization of the classical Plemelj-Privalov theorem, identifying a large class of singular integral operators that are bounded on generalized Hölder spaces. This class includes the Cauchy-Clifford operator and the harmonic double layer operator, among others. |
| title | Characterizations of Lyapunov domains in terms of Riesz transforms and the Plemelj-Privalov theorem |
| topic | Classical Analysis and ODEs 26A16, 28A75, 31B10, 42B20, 42B35, 42B37, 15A66 |
| url | https://arxiv.org/abs/2604.19143 |