Jihun Yum, Characterization of Diederich-Fornaess and Steinness Indices in Cⁿ

Let Ω be a bounded pseudoconvex domain in Cⁿ with smooth boundary ∂Ω. The Diederich-Fornaess index and the Steinness index of Ω are defined by

DF(Ω) := sup_ρ { 0 < η < 1 : -(-ρ)^η is strictly plurisubharmonic on Ω ∩ U for some neighborhood U of ∂Ω },

S(Ω) := inf_ρ { η > 1 : ρ^η is strictly plurisubharmonic on Ω^c ∩ U for some neighborhood U of ∂Ω },

where ρ is a defining function for Ω.

First, we see basic properties and known results about the Diederich-Fornaess and Steinness indices. Also, we see the relation between two indices on a 1-parameter family of domains in C², called worm domains, constructed by Diederich and Fornaess.