Boundary invariants of pseudoconvex domains. (English) Zbl 0583.32048

Catlin, David

Ann. Math. (2) 120, 529-586 (1984).

Let \(\Omega \subseteq {\mathbb{C}}^ n\) be a smoothly bounded pseudoconvex domain. A notion of multitype of a point \(P\in \partial \Omega\) is introduced. This term is defined in terms of directional derivatives of a defining function for \(\partial \Omega\). The principal theorem of this paper demonstrates the naturality of the notion of multitype. In particular, the concept of multitype is compared to an algebro-geometric notion which was introduced previously by D’Angelo.
In a later paper (to appear) Catlin proves that his notion of finite multitype is equivalent with the existence of subelliptic estimates for the \({\bar \partial}\)-Neumann problem.

Reviewer: St.Krantz

MSC:

32T99	Pseudoconvex domains
32W05	\(\overline\partial\) and \(\overline\partial\)-Neumann operators
35N15	\(\overline\partial\)-Neumann problems and formal complexes in context of PDEs

Keywords:

multitype of boundary points; boundary invariant; smoothly bounded pseudoconvex domain; existence of subelliptic estimates for the \({\bar \partial }\)-Neumann problem