$L^2$-Sobolev Theory for $\bar\partial$ on Domains in $\Bbb {CP}^n$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911079164018688 |
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| author | Shaw, Mei-Chi |
| author_facet | Shaw, Mei-Chi |
| contents | In this article, we study the range of the Cauchy-Riemann operator $\bar\partial$ on domains in the complex projective space $\Bbb{CP}^n$. In particular, we show that $\bar\partial$ does not have closed range in $L^2$ for (2,1)-forms on the Hartogs triangle in $\Bbb{CP}^2$. We also study the $\bar\partial$-Cauchy problem on pseudoconvex domains and use it to prove the Sobolev estimates for $\bar\partial$ on pseudoconcave domains in $\Bbb{CP}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19355 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^2$-Sobolev Theory for $\bar\partial$ on Domains in $\Bbb {CP}^n$ Shaw, Mei-Chi Complex Variables 32W05, 35N15 In this article, we study the range of the Cauchy-Riemann operator $\bar\partial$ on domains in the complex projective space $\Bbb{CP}^n$. In particular, we show that $\bar\partial$ does not have closed range in $L^2$ for (2,1)-forms on the Hartogs triangle in $\Bbb{CP}^2$. We also study the $\bar\partial$-Cauchy problem on pseudoconvex domains and use it to prove the Sobolev estimates for $\bar\partial$ on pseudoconcave domains in $\Bbb{CP}^n$. |
| title | $L^2$-Sobolev Theory for $\bar\partial$ on Domains in $\Bbb {CP}^n$ |
| topic | Complex Variables 32W05, 35N15 |
| url | https://arxiv.org/abs/2507.19355 |