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

Abstract

We develop a method for proving sup-norm and Hölder 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 Hölder estimates for all bounded, pseudoconvex domains with real-analytic boundary.

29 pages