Joint KASS and Analysis and Probability Seminar

A domain Ω ⊂ ℂn is said to be pseudoconvex if − log(−δ(z)) is plurisubharmonic in Ω, where δ is a signed distance function of Ω. The study of global regularity of ∂¯-Neumann problem on bounded pseudoconvex domains is dated back to the 1960s. However, a complete understanding of the regularity is still absent. On the other hand, the Diederich–Fornæss index was introduced in 1977 originally for seeking bounded plurisubharmonic functions. Through decades, enormous evidence has indicated a relationship between global regularity of the ∂¯-Neumann problem and the Diederich–Fornæss index. Indeed, it has been a long-lasting open question whether the trivial Diederich–Fornæss index implies global regularity. In this talk, we will introduce the backgrounds and motivations. The main theorem of the talk proved recently by Emil Straube and me answers this open question for (0, n − 1) forms.