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

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. 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.

arXiv comment70 pages. Some errors in earlier version have been fixed along with some reorganization