Necessary and sufficient conditions for global regularity of the ∂̄-Neumann problem

Determine necessary and sufficient conditions for global regularity of the \bar\partial-Neumann problem, in particular to characterize precisely when the \bar\partial-Neumann operator exhibits global regularity on bounded pseudoconvex domains in \mathbb{C}^n endowed with the induced Euclidean metric.

Background

The paper contrasts analysis on complete manifolds with the case of bounded domains in Euclidean space, noting that the latter (incomplete metrics) is less understood. Within this context, the \bar\partial-Neumann problem is highlighted as sharing the same lack of clarity, especially regarding global regularity.

The authors reference several works that provide sufficient conditions for global regularity, underscoring that despite substantial progress, a full characterization remains elusive. This motivates their development of new L2 estimates applicable beyond strictly plurisubharmonic weights.

References

Indeed, the necessary and sufficient conditions of the global regularity of the $\bar\partial$-Neumann problem remain unknown.

— On the Hörmander's estimate  (2401.16573 - Liu, 2024) in Section 1 (Introduction)

It is still unclear if and how it impacts the actual boundary behavior of holomorphic mappings between smooth bounded pseudoconvex domains.

— Obstructions to exact regularity of sublaplacians  (2610.03037 - Dall'Ara, 2 Oct 2026) in Section 1.3, “Motivations and antecedents,” Introduction