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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2512.01787 |
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| _version_ | 1866911609948995584 |
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| author | Li, Yong Zhang, Yuchen |
| author_facet | Li, Yong Zhang, Yuchen |
| contents | In this paper, we introduce a pair of multiplication-like operations, $L_0$ and $L_1$, which derive $k$-regular functions from $(k+1)$-regular functions. The investigation of the inverse problem naturally leads to a deeper study of the 2-Cauchy-Fueter equation. In doing so, we provide a new acyclic resolution for the sheaf of $2$-regular functions $\mathcal{R}^{(2)}$. Furthermore, a complete topological characterization for the solvability of the $2$-Cauchy-Fueter equation is established. Specifically, we prove that the $2$-Cauchy-Fueter equation $$\mathscr{D}^{(2)}f=g$$ is solvable for any $g$ satisfying $\mathscr{D}_1^{(2)}g=0$ on a domain $Ω\subset\mathbb{R}^4$ if and only if $H^3(Ω, \mathbb{R}) = 0$, or equivalently, if and only if every real-valued harmonic function on $Ω$ can be represented as the real part of a quaternionic regular function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01787 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Pair of Multiplication-Type Operators in Quaternionic Analysis and the 2-Cauchy-Fueter Equation Li, Yong Zhang, Yuchen Complex Variables In this paper, we introduce a pair of multiplication-like operations, $L_0$ and $L_1$, which derive $k$-regular functions from $(k+1)$-regular functions. The investigation of the inverse problem naturally leads to a deeper study of the 2-Cauchy-Fueter equation. In doing so, we provide a new acyclic resolution for the sheaf of $2$-regular functions $\mathcal{R}^{(2)}$. Furthermore, a complete topological characterization for the solvability of the $2$-Cauchy-Fueter equation is established. Specifically, we prove that the $2$-Cauchy-Fueter equation $$\mathscr{D}^{(2)}f=g$$ is solvable for any $g$ satisfying $\mathscr{D}_1^{(2)}g=0$ on a domain $Ω\subset\mathbb{R}^4$ if and only if $H^3(Ω, \mathbb{R}) = 0$, or equivalently, if and only if every real-valued harmonic function on $Ω$ can be represented as the real part of a quaternionic regular function. |
| title | A Pair of Multiplication-Type Operators in Quaternionic Analysis and the 2-Cauchy-Fueter Equation |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2512.01787 |