Regularity in the $\overline{\partial}$--Neumann problem, D'Angelo forms, and Diederich--Fornæss index
Emil J. Straube
TL;DR
This survey links the regularity of the $\overline{\partial}$-Neumann problem to D'Angelo forms and the Diederich–Fornæss index, tracing developments from Boas–Straube’s early Sobolev regularity results to Kohn’s quantitative index program and subsequent refinements by Harrington, Liu, and Yum. A central theme is expressing commutator control via $\alpha_{\rho}$ and showing that approximate exactness or cohomological vanishing of these forms on Levi-null directions yields Sobolev regularity, with index-one domains conjectured to exhibit global regularity. The DF–index is reformulated in terms of differential inequalities for D'Angelo forms, enabling a precise boundary-based criterion and connecting geometric-type conditions with functional-analytic regularity results. Recent work by Liu and Liu–Straube shows that, under comparable Levi eigenvalue sums, index one implies Sobolev regularity without extra geometric hypotheses, at least in low dimensions, aligning with the long-standing conjecture and highlighting the deep interaction between complex geometry and PDE regularity.
Abstract
This article chronicles a development that started around 1990 with \cite{BoasStraube91}, where the authors showed that if a smooth bounded pseudoconvex domain $Ω$ in $\mathbb{C}^{n}$ admits a defining function that is plurisubharmonic at points of the boundary, then the $\overline{\partial}$--Neumann operators on $Ω$ preserve the Sobolev spaces $W^{s}_{(0,q)}(Ω)$, $s\geq 0$. 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--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 $\overline{\partial}$--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.