Topics in Function Theory on Strictly Pseudoconvex Domains

Abstract

Every domain in the complex plane with C ² boundary is strictly (Levi) pseudoconvex. In this chapter we generalize several classical function theoretic results from planar domains to strictly pseudoconvex domains in ℂⁿ. In contrast to the results on arbitrary domains of holomorphy discussed in Chapter VI, the emphasis here will be on the behavior of holomorphic functions and other analytic objects up to the boundary of the domain. In somewhat more detail, we will present the construction and basic properties of two analogues of the Cauchy kernel for a strictly pseudoconvex domain D, and of a solution operator for ∂̅ on D with L ^P estimates for 1 ≤ p ≤ ∞. Moreover, we will discuss applications of these results to uniform and L ^p approximation by holomorphic functions and to ideals in the algebra A(D) of holomorphic functions with continuous boundary values. The highlight will be a regularity theorem for the Bergman projection based on a rather explicit representation of the abstract Bergman kernel, and its application to the study of boundary regularity of biholomorphic maps.