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 \(\Omega\) in \(\mathbb{CP}^n\) there exists a Lipschitz defining function \(\rho\) and an exponent \(0<\eta<1\) such that \(-(-\rho)^\eta\) is strictly plurisubharmonic on \(\Omega\). 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 \(\rho=-\delta\), where \(\delta\) is the geodesic distance function for the boundary of \(\Omega\).