Sup-norm Estimates for $\overline{\partial}$ in $\mathbb{C}^3$

TL;DR

This paper introduces a new method to establish sup-norm and Hölder estimates for the $ar{ ext{d}}$ operator on broad classes of finite type pseudoconvex domains in $ extbf{C}^3$, overcoming previous obstructions posed by complex curves with high contact order.

Contribution

It provides a novel approach to handle singular complex curves with high boundary contact, proving sup-norm and Hölder estimates for all bounded, real-analytic boundary pseudoconvex domains in $ extbf{C}^3$.

Findings

Established sup-norm estimates for $ar{ ext{d}}$ in $ extbf{C}^3$ domains.

Proved Hölder estimates for these domains.

Addressed the obstacle of high contact order complex curves.

Abstract

We develop a method for proving sup-norm and H\"older estimates for $\overline{\partial}$ on wide class of finite type pseudoconvex domains in $\mathbb{C}^n$. A fundamental obstruction to proving sup-norm estimates is the possibility of singular complex curves with exceptionally high order of contact with the boundary. Our method handles this problem, and in $\mathbb{C}^3$, we prove sup-norm and H\"older estimates for all bounded, pseudoconvex domains with real-analytic boundary.