2025/04/04 by Emil J. Sträube, Straube, Emil J.
Mathematics · #32T99 #32W05 #Advanced Algebra and Geometry #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2504.03562
openalex publication_date 2025/04/04 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
This article chronicles a development that started around 1990 with \citeBoasStraube91, where the authors showed that if a smooth bounded pseudoconvex domain Ω in ℂn admits a defining function that is plurisubharmonic at points of the boundary, then the ∂--Neumann operators on Ω preserve the Sobolev spaces Ws(0,q)(Ω), s≥ 0. The same authors then proved a further regularity result and made explicit the role of D'Angelo forms for regularity (\citeBoasStraube93). A few years later, Kohn (\citeKohn99) initiated a quantitative study of the results in \citeBoasStraube91 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 (\citeHarrington11,Harrington19,Harrington22). Then, around 2020, Liu (\citeLiu19b, Liu19) and Yum (\citeYum21) 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 ∂--Neumann problem (\citeLiuStraube22). Much of the work described above relies heavily on Kohn's groundbreaking contributions to the regularity theory of the ∂--Neumann problem.