# On Competing Definitions for the Diederich-Forn{\ae}ss Index

## TL;DR

This paper investigates different definitions of the Diederich-Forn{
a}ss index for bounded pseudoconvex domains, establishing equivalences among various families of functions used in its computation.

## Contribution

It proves that the Diederich-Forn{
a}ss index remains the same when computed with upper semi-continuous or Lipschitz functions, and with $C^k$ or $C^2$ functions for sufficiently smooth boundaries.

## Key findings

- Equivalence of the index for upper semi-continuous and Lipschitz functions.
- Equivalence of the index for $C^k$ and $C^2$ functions when boundary is $C^k$.
- Provides a unified understanding of the Diederich-Forn{
a}ss index definitions.

## Abstract

Let $\Omega\subset\mathbb{C}^n$ be a bounded pseudoconvex domain. We define the Diederich-Forn{\ae}ss 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-Forn{\ae}ss index with respect to the family of upper semi-continuous functions is the same as computing the Diederich-Forn{\ae}ss index with respect to the family of Lipschitz functions. When the boundary of $\Omega$ is $C^k$, $k\geq 2$, we prove that the Diederich-Forn{\ae}ss index with respect to the family of $C^k$ functions is the same as the Diederich-Forn{\ae}ss index with respect to the family of $C^2$ functions.