Estimates for the $\bar{\partial}$-equation on canonical surfaces
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2018
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| Acceso en línea: | |
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| _version_ | 1866913597937942528 |
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| author | Andersson, Mats Lärkäng, Richard Ruppenthal, Jean Kalm, Håkan Samuelsson Wulcan, Elizabeth |
| author_facet | Andersson, Mats Lärkäng, Richard Ruppenthal, Jean Kalm, Håkan Samuelsson Wulcan, Elizabeth |
| contents | We study the solvability in $L^p$ of the $\bar\partial$-equation in a neighborhood of a canonical singularity on a complex surface, a so-called du Val singularity. We get a quite complete picture in case $p=2$ for two natural closed extensions $\bar\partial_s$ and $\bar\partial_w$ of $\bar\partial$. For $\bar\partial_s$ we have solvability, whereas for $\bar\partial_w$ there is solvability if and only if a certain boundary condition $(*)$ is fulfilled at the singularity. Our main tool is certain integral operators for solving $\bar\partial$ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_01004 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Estimates for the $\bar{\partial}$-equation on canonical surfaces Andersson, Mats Lärkäng, Richard Ruppenthal, Jean Kalm, Håkan Samuelsson Wulcan, Elizabeth Complex Variables We study the solvability in $L^p$ of the $\bar\partial$-equation in a neighborhood of a canonical singularity on a complex surface, a so-called du Val singularity. We get a quite complete picture in case $p=2$ for two natural closed extensions $\bar\partial_s$ and $\bar\partial_w$ of $\bar\partial$. For $\bar\partial_s$ we have solvability, whereas for $\bar\partial_w$ there is solvability if and only if a certain boundary condition $(*)$ is fulfilled at the singularity. Our main tool is certain integral operators for solving $\bar\partial$ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity. |
| title | Estimates for the $\bar{\partial}$-equation on canonical surfaces |
| topic | Complex Variables |
| url | https://arxiv.org/abs/1804.01004 |