Sobolev and Hölder estimates for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910969339314176 |
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| author | Shi, Ziming |
| author_facet | Shi, Ziming |
| contents | We prove a homotopy formula which yields almost sharp estimates in all (positive-indexed) Sobolev and Hölder-Zygmund spaces for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$, extending the earlier results of Fefferman-Kohn (1988), Range (1990), and Chang-Nagel-Stein (1992). The main novelty of our proof is the construction of holomorphic support functions that admit precise estimates when the parameter variable lies in a thin shell outside the domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07655 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sobolev and Hölder estimates for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$ Shi, Ziming Complex Variables Analysis of PDEs We prove a homotopy formula which yields almost sharp estimates in all (positive-indexed) Sobolev and Hölder-Zygmund spaces for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$, extending the earlier results of Fefferman-Kohn (1988), Range (1990), and Chang-Nagel-Stein (1992). The main novelty of our proof is the construction of holomorphic support functions that admit precise estimates when the parameter variable lies in a thin shell outside the domain. |
| title | Sobolev and Hölder estimates for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$ |
| topic | Complex Variables Analysis of PDEs |
| url | https://arxiv.org/abs/2410.07655 |