Bounded Plurisubharmonic Exhaustion Functions for Lipschitz Pseudoconvex Domains in $\mathbb{CP}^n$

Abstract

In this paper, we use Takeuchi's Theorem to show that for every Lipschitz pseudoconvex domain $Ω$ in $\mathbb{CP}^n$ there exists a Lipschitz defining function $ρ$ and an exponent $0<η<1$ such that $-(-ρ)^η$ is strictly plurisubharmonic on $Ω$. This generalizes a result of Ohsawa and Sibony for $C^2$ domains. In contrast to the Ohsawa-Sibony result, we provide a counterexample demonstrating that we may not assume $ρ=-δ$, where $δ$ is the geodesic distance function for the boundary of $Ω$.