On competing definitions for the Diederich–Fornæss index

Abstract

Let \(\Omega \subset {\mathbb {C}}^n\) be a bounded pseudoconvex domain. We define the Diederich–Fornæss index with respect to a family of functions to be the supremum over the set of all exponents \(0<\eta <1\) such that there exists a function \(\rho _\eta \) in this family that is comparable to \(-{{\,\mathrm{dist}\,}}(z,b\Omega )\) on \(\Omega \) and such that \(-(-\rho _\eta )^\eta \) is plurisubharmonic on \(\Omega \). We first prove that computing the Diederich–Fornæss index with respect to the family of upper semi-continuous functions is the same as computing the Diederich–Fornæss index with respect to the family of Lipschitz functions. When the boundary of \(\Omega \) is \(C^k\), \(k\ge 2\), we prove that the Diederich–Fornæss index with respect to the family of \(C^k\) functions is the same as the Diederich–Fornæss index with respect to the family of \(C^2\) functions.