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.
Categories: math.CV