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

TL;DR

This paper shows that failures in $L^1$ regularity within Calderón-Zygmund theory are universal, using holomorphic functions to generate counterexamples to the Poisson equation, linking harmonic analysis with complex geometry.

Contribution

It introduces sharp level-set estimates connecting harmonic analysis and complex geometry, providing universal counterexamples to Calderón-Zygmund theory.

Findings

Holomorphic functions generate counterexamples to the Poisson equation.

Sharp level-set estimates link harmonic analysis with complex geometry.

Failure of $L^1$ regularity is shown to be universal.

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.