Boundary Invariants of Pseudoconvex Domains

Abstract

The purpose of this paper is to describe an invariant AX(z) which is associated with each point z in the boundary of a smoothly bounded pseudoconvex domain in Cn. The motivation for defining the invariant AX(z), referred to as the multitype of z, comes from the study of the boundary regularity properties of solutions of the d-Neumann problem on domains of finite type. By a domain of finite type, we mean one that satisfies the definition given by D'Angelo [5], namely a domain such that at any boundary point the maximum order of contact of one-dimensional complex-analytic varieties with the boundary is bounded. In [4] the author showed how one can give a proof of global regularity of the a-Neumann problem for pseudoconvex domains of finite type if there is an invariant such as the one described in this paper. In a forthcoming article this invariant will be used to obtain a proof of subelliptic estimates for the same class of domains. Before giving the definition of the multitype AX(z), we must introduce some