Pseudoconvex Domains II —Solution

In the previous chapter the Pseudoconvexity Problem is reduced to the problem of asking if a pseudoconvex domain is Stein. In this chapter we solve it affirmatively. It is the high point to prove that a bounded domain with strongly pseudoconvex boundary is Stein (Levi’s Problem). We shall give two proofs to it; the first is K. Oka’s original one due to an unpublished paper of 1943 by means of the Fredholm integral equation of the second kind type combined with the Joku-Iko Principle, and the second is due to H. Grauert (1958) through L. Schwartz’s Fredholm Theorem for compact operators and the bumping method. The comparison is interesting. Each proof has its own advantage.