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 such that $-{\rho }_{\eta }$ is comparable to the distance to the boundary of $\Omega$ 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.