Energy asymptotics of holomorphic functions with application to Calderón-Zygmund theory in ℂ
Abstract
The Calderón-Zygmund theory establishes the boundedness of singular integral operators on Lp spaces for 1 < p < ∞, yet it encounters a failure at the endpoint p = 1. While radial counterexamples in ℝn 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 ℂ. Additionally, we construct a new family of counterexamples at the endpoint p = ∞, showing that the failure of W2,∞-regularity is also a universal phenomenon in complex one dimension.
Related
- Wavelet resolution and Sobolev regularity of Calderón–Zygmund operators on domains
- Sparse bounds for rough non-translation-invariant Calderón-Zygmund operators
- A Morrey-to-Lebesgue Equivalence for Convolution Calderón--Zygmund Operators
- Sharp Matrix-Weighted Lower Bounds for Non-Degenerate Convolution Calderón--Zygmund Operators
- BMO Domains and Quasicircles Associated with Zygmund Functions
BibTeX
@misc{chen2026energy,
title = {{Energy asymptotics of holomorphic functions with application to Calderón-Zygmund theory in $\mathbbC$}},
author = {Hongrong Chen and Guokuan Shao and Jujie Wu and Wei Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02601},
eprint = {2609.02601},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02601},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Energy asymptotics of holomorphic functions with application to Calderón-Zygmund theory in ℂ
AU - Hongrong Chen
AU - Guokuan Shao
AU - Jujie Wu
AU - Wei Xia
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02601
UR - https://arxiv.org/abs/2609.02601
AB - The Calderón-Zygmund theory establishes the boundedness of singular integral operators on L^p spaces for 1 < p < ∞, yet it encounters a failure at the endpoint p = 1. While radial counterexamples in 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 ℂ. Additionally, we construct a new family of counterexamples at the endpoint p = ∞, showing that the failure of W^2,∞-regularity is also a universal phenomenon in complex one dimension.
ER -