Regularity in the $$\overline{\partial }$$–Neumann Problem, D’Angelo Forms, and Diederich–Fornæss Index
This article chronicles a development that started around 1990 with [17], where the authors showed that if a smooth bounded pseudoconvex domain $$\Omega $$ Ω in $$\mathbb {C}^{n}$$ C n admits a defining function that is plurisubharmonic at points of the boundary, then the $$\overline{\partial }$$ ∂...
Saved in:
| Published in: | The Journal of geometric analysis Vol. 35; no. 10 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
01.10.2025
|
| ISSN: | 1050-6926, 1559-002X |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | This article chronicles a development that started around 1990 with [17], where the authors showed that if a smooth bounded pseudoconvex domain $$\Omega $$ Ω in $$\mathbb {C}^{n}$$ C n admits a defining function that is plurisubharmonic at points of the boundary, then the $$\overline{\partial }$$ ∂ ¯ –Neumann operators on $$\Omega $$ Ω preserve the Sobolev spaces $$W^{s}_{(0,q)}(\Omega )$$ W ( 0 , q ) s ( Ω ) , $$s\ge 0$$ s ≥ 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. |
|---|---|
| ISSN: | 1050-6926 1559-002X |
| DOI: | 10.1007/s12220-025-02120-2 |