Degenerate quarternionic Monge-Ampère equations in weighted energy classes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915262807146496 |
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| author | Lin, Genglong |
| author_facet | Lin, Genglong |
| contents | In this paper, we consider degenerate quaternionic Monge-Ampère equations in weighted energy class $\mathcal{E}_χ(Ω)$ where $Ω$ is a quarternionic domain in $\mathbb{H}^n$ and $χ$ is a weight function which satisfies some natural conditions. Firstly we prove that the quaternionic Monge-Ampère operator is well-defined for functions in $\mathcal{E}_χ(Ω)$, in particular $\mathcal{E}_p(Ω),p>0$.
Secondly, we prove that fine property holds in the Cegrell type class $\mathcal{E}(Ω)$. As an application, we prove a mass concentration theorem for the quarternionic plurisubharmonic envelope.
In the study of complex Monge-Ampère equation, characterization of finite energy range of complex Monge-Ampère operator was a central problem which aroused the interest of experts in the subject. As a quaternionic analogue, we prove a theorem which explicitly characterizes the finite energy range of \emph{quaternionic} Monge-Ampère operator in the end. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19619 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Degenerate quarternionic Monge-Ampère equations in weighted energy classes Lin, Genglong Complex Variables Analysis of PDEs Differential Geometry 32U15, 31C10, 32U40 In this paper, we consider degenerate quaternionic Monge-Ampère equations in weighted energy class $\mathcal{E}_χ(Ω)$ where $Ω$ is a quarternionic domain in $\mathbb{H}^n$ and $χ$ is a weight function which satisfies some natural conditions. Firstly we prove that the quaternionic Monge-Ampère operator is well-defined for functions in $\mathcal{E}_χ(Ω)$, in particular $\mathcal{E}_p(Ω),p>0$. Secondly, we prove that fine property holds in the Cegrell type class $\mathcal{E}(Ω)$. As an application, we prove a mass concentration theorem for the quarternionic plurisubharmonic envelope. In the study of complex Monge-Ampère equation, characterization of finite energy range of complex Monge-Ampère operator was a central problem which aroused the interest of experts in the subject. As a quaternionic analogue, we prove a theorem which explicitly characterizes the finite energy range of \emph{quaternionic} Monge-Ampère operator in the end. |
| title | Degenerate quarternionic Monge-Ampère equations in weighted energy classes |
| topic | Complex Variables Analysis of PDEs Differential Geometry 32U15, 31C10, 32U40 |
| url | https://arxiv.org/abs/2504.19619 |