Energy asymptotics of holomorphic functions with application to Calderón-Zygmund theory in $\mathbb{C}$

Abstract

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.