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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.01559 |
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| _version_ | 1866913079106732032 |
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| author | Pan, Yifei Shao, Guokuan Wang, Jianfei Wu, Jujie |
| author_facet | Pan, Yifei Shao, Guokuan Wang, Jianfei Wu, Jujie |
| contents | We demonstrate that the failure of $L^1$ regularity in Calderón-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in $\C^n$ generates a counterexample to the Poisson equation.
In order to achieve this goal, we shall establish sharp level-set estimates that link harmonic analysis to the geometry of complex structure through Hironaka's resolution of singularities and the Łojasiewicz gradient inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01559 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Energy estimates for level sets of holomorphic functions and universal counterexamples to Calderón-Zygmund theory Pan, Yifei Shao, Guokuan Wang, Jianfei Wu, Jujie Complex Variables Analysis of PDEs 32A10, 42B37, 42B20 We demonstrate that the failure of $L^1$ regularity in Calderón-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in $\C^n$ generates a counterexample to the Poisson equation. In order to achieve this goal, we shall establish sharp level-set estimates that link harmonic analysis to the geometry of complex structure through Hironaka's resolution of singularities and the Łojasiewicz gradient inequality. |
| title | Energy estimates for level sets of holomorphic functions and universal counterexamples to Calderón-Zygmund theory |
| topic | Complex Variables Analysis of PDEs 32A10, 42B37, 42B20 |
| url | https://arxiv.org/abs/2604.01559 |