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

We demonstrate that the failure of $L^1$ regularity in Calderón-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in ${\text{C}}^n$ 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 Łojasiewicz gradient inequality.