Domains of finite type and Hölder continuity of the Perron-Bremermann function

Abstract:

Let $\Omega$ be a smoothly bounded domain in ${\mathbb C}^n$ such that $0\in \partial \Omega$. We give a bound for the type of $\partial \Omega$ at 0 in terms of the Hölder exponent of its Perron-Bremermann function with simple boundary data. We then use this to show that a smoothly bounded domain in ${\mathbb C}^2$ is pseudoconvex of finite type if and only if its Perron-Bremermann function corresponding to Hölder continuous boundary data is Hölder continuous.