This article chronicles a development that started around 1990 with [17], 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 \(W^{s}_{(0,q)}(\Omega )\) , \(s\ge 0\) . The same authors then proved a further regularity result and made explicit the role of D’Angelo forms for regularity [19]. A few years later, Kohn [69] initiated a quantitative study of the results in [17] 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 [51–53]. Then, around 2020, Liu [72, 73] and Yum [101] 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 [75]. Much of the work described above relies heavily on Kohn’s groundbreaking contributions to the regularity theory of the \(\overline{\partial }\) –Neumann problem.
