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Energy estimates for level sets of holomorphic functions and counterexamples to Calderón-Zygmund theory

Yifei Pan, Guokuan Shao, Jianfei Wang, Jujie Wu

Abstract

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. Using Hironaka's resolution of singularities and the Łojasiewicz gradient inequality, we establish sharp level-set estimates that link harmonic analysis to the geometry of complex structure, providing results of independent interest in algebraic geometry.

Energy estimates for level sets of holomorphic functions and counterexamples to Calderón-Zygmund theory

Abstract

We demonstrate that the failure of regularity in Calderón-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in generates a counterexample to the Poisson equation. Using Hironaka's resolution of singularities and the Łojasiewicz gradient inequality, we establish sharp level-set estimates that link harmonic analysis to the geometry of complex structure, providing results of independent interest in algebraic geometry.

Paper Structure

This paper contains 21 sections, 31 theorems, 347 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Energy and volume estimates for (sub)level sets of holomorphic functions
  3. Regularity of small level sets
  4. The coarea formula and Dirac delta function
  5. Energy estimates for the case of monomials
  6. Case $n = 2$ for $I(\varepsilon)$
  7. Case $n=3$ for $I(\varepsilon)$
  8. Case $n=2$ for $J(\varepsilon)$
  9. Continuity of $I(\varepsilon)$
  10. Proof of Theorem \ref{['t02']}
  11. Proof of Theorem \ref{['tC']}
  12. Counterexamples to Calderón-Zygmund theory
  13. Elementary computations for $u$ and $v$
  14. Proof of Theorem \ref{['t01']}
  15. Proof of Theorem \ref{['th:maintheorem2']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $U$ be the unit polydisc in $\mathbb{C}^{n}$ and $f(z)$ a holomorphic function in a neighborhood of $\overline{U}$ with the non empty zero set $Z(f)$. Then where $dS$ is the surface measure on the level set and $dV$ is the Lebesgue measure. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Tubular neighbourhood in the unit ball $\mathbb{B}^n$

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Regularity of small level sets
  • proof
  • Lemma 2.4
  • ...and 54 more