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arXiv · 2604.01559

Energy estimates for level sets of holomorphic functions and universal counterexamples to Calder\'on-Zygmund theory

Yifei Pan·Guokuan Shao·Jianfei Wang·Jujie Wu
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Abstract

We demonstrate that the failure of $L^1$ regularity in Calder\'on-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 \L{}ojasiewicz gradient inequality.

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Yifei Pan, Guokuan Shao, Jianfei Wang, Jujie Wu. 2026-04-02. Energy estimates for level sets of holomorphic functions and universal counterexamples to Calder\'on-Zygmund theory. https://arxiv.org/abs/2604.01559

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