Regularity in the $\bar{\mathrm{\partial }}$--Neumann problem, D'Angelo forms, and Diederich--Fornæss index

This article chronicles a development that started around 1990 with \cite{BoasStraube91}, where the authors showed that if a smooth bounded pseudoconvex domain $\mathrm{\Omega }$ in ${\mathbb{C}}^n$ admits a defining function that is plurisubharmonic at points of the boundary, then the $\bar{\mathrm{\partial }}$–Neumann operators on $\mathrm{\Omega }$ preserve the Sobolev spaces $W_{(0,q)}^s(\mathrm{\Omega })$, $s\ge 0$. The same authors then proved a further regularity result and made explicit the role of D’Angelo forms for regularity (\cite{BoasStraube93}). A few years later, Kohn (\cite{Kohn99}) initiated a quantitative study of the results in \cite{BoasStraube91} by relating the Sobolev level up to which regularity holds to the Diederich–Fornæss index of the domain. Many of these ideas were synthesized and developed further by Harrington (\cite{Harrington11,Harrington19,Harrington22}). Then, around 2020, Liu (\cite{Liu19b, Liu19}) and Yum (\cite{Yum21}) discovered that the DF–index is closely related to certain differential inequalities involving D’Angelo forms. This relationship in turn led to a recent new result which supports the conjecture that DF–index one should imply global regularity in the $\bar{\mathrm{\partial }}$–Neumann problem (\cite{LiuStraube22}). Much of the work described above relies heavily on Kohn’s groundbreaking contributions to the regularity theory of the $\bar{\mathrm{\partial }}$–Neumann problem.