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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2205.12025 |
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| _version_ | 1866910588741877760 |
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| author | Gong, Xianghong |
| author_facet | Gong, Xianghong |
| contents | We study regularity of solutions $u$ to $\overline\partial u=f$ on a relatively compact $C^2$ domain $D$ in a complex manifold of dimension $n$, where $f$ is a $(0,q)$ form. Assume that there are either $(q+1)$ negative or $(n-q)$ positive Levi eigenvalues at each point of boundary $\partial D$. Under the necessary condition that a locally $L^2$ solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain $1/2$ derivative when $q=1$ and $f$ is in the Hölder-Zygmund space $Λ^r(\overline D)$ with $r>1$. For $q>1$, the same regularity for the solutions is achieved when $\partial D$ is either sufficiently smooth or of $(n-q)$ positive Levi eigenvalues everywhere on $\partial D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_12025 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On regularity of $\overline\partial$-solutions on $a_q$ domains with $C^2$ boundary in complex manifolds Gong, Xianghong Complex Variables We study regularity of solutions $u$ to $\overline\partial u=f$ on a relatively compact $C^2$ domain $D$ in a complex manifold of dimension $n$, where $f$ is a $(0,q)$ form. Assume that there are either $(q+1)$ negative or $(n-q)$ positive Levi eigenvalues at each point of boundary $\partial D$. Under the necessary condition that a locally $L^2$ solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain $1/2$ derivative when $q=1$ and $f$ is in the Hölder-Zygmund space $Λ^r(\overline D)$ with $r>1$. For $q>1$, the same regularity for the solutions is achieved when $\partial D$ is either sufficiently smooth or of $(n-q)$ positive Levi eigenvalues everywhere on $\partial D$. |
| title | On regularity of $\overline\partial$-solutions on $a_q$ domains with $C^2$ boundary in complex manifolds |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2205.12025 |