AI 中文总结
该研究针对复一维情形，推导了全纯函数水平集积分与次水平集能量的带显式常数的渐近公式，简化了p=1端点处Calderón-Zygmund理论普适反例的证明，还构造了p=∞端点处的新反例族，证明W^{2,∞}正则性失效是复一维情形的普适现象。
AI 中文摘要
Calderón-Zygmund理论确立了奇异积分算子在1<p<∞时于L^p空间上的有界性，但在端点p=1处失效。虽然实空间R^n中的径向反例已被充分记录，Pan-Shao-Wang-Wu等人[psww2026]证明，每个非常数全纯函数都为Calderón-Zygmund框架下的泊松方程提供了反例，其奇异轨迹为余维1的复子簇。本文聚焦复一维情形，建立了更强的结果：我们证明了水平集积分和次水平集能量带有显式常数的渐近公式；随后给出复平面上Calderón-Zygmund理论在p=1处普适反例的简化证明；此外，我们构造了p=∞端点处的新反例族，表明W^{2,∞}正则性的失效在复一维情形也是一种普适现象。
英文摘要
The Calderón-Zygmund theory establishes the boundedness of singular integral operators on $L^p$ spaces for $1 < p < \infty$, yet it encounters a failure at the endpoint $p = 1$. While radial counterexamples in $\mathbb{R}^n$ are well-documented, Pan-Shao-Wang-Wu \cite{psww2026} has showed that every nonconstant holomorphic function provides a counterexample to the Poisson equation within the Calderón-Zygmund framework, with the singular locus being a complex subvariety of codimension one. In this paper, we focus on the complex one-dimensional case and establish stronger results. We prove asymptotic formulas with explicit constants for both the level-set integral and the sublevel-set energy. Then we give simplified proofs of the universal counterexamples to Calderón-Zygmund theory at $p = 1$ in $\mathbb{C}$. Additionally, we construct a new family of counterexamples at the endpoint $p = \infty$, showing that the failure of $W^{2,\infty}$-regularity is also a universal phenomenon in complex one dimension.
Comments23 pages
