Regularity in the $\overline{\partial}$--Neumann problem, D'Angelo forms, and Diederich--Forn\ae ss index

Abstract

This article chronicles a development that started around 1990 with \cite{BoasStraube91}, where the authors showed that if a smooth bounded pseudoconvex domain $\Omega$ in $\mathbb{C}^{n}$ admits a defining function that is plurisubharmonic at points of the boundary, then the $\overline{\partial}$--Neumann operators on $\Omega$ preserve the Sobolev spaces $W^{s}_{(0,q)}(\Omega)$, $s\geq 0$. 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--Forn\ae ss 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 $\overline{\partial}$--Neumann problem (\cite{LiuStraube22}). Much of the work described above relies heavily on Kohn's groundbreaking contributions to the regularity theory of the $\overline{\partial}$--Neumann problem.