Regularity in the $$\overline{\partial }$$–Neumann Problem, D’Angelo Forms, and Diederich–Fornæss Index

Abstract This article chronicles a development that started around 1990 with [17], where the authors showed that if a smooth bounded pseudoconvex domain $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> in $$\mathbb {C}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> admits a defining function that is plurisubharmonic at points of the boundary, then the $$\overline{\partial }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>∂</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> –Neumann operators on $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> preserve the Sobolev spaces $$W^{s}_{(0,q)}(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , $$s\ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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 $$\overline{\partial }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>∂</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> –Neumann problem [75]. Much of the work described above relies heavily on Kohn’s groundbreaking contributions to the regularity theory of the $$\overline{\partial }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>∂</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> –Neumann problem.