ABSTRACT
Given a continuous function ϕ on bΩ, the classical Dirichlet problem is to find a harmonic function u on Ω that extends continuously to bΩ and that agrees with ϕ on bΩ. In this chapter, we will relate the solution of this problem to the Szego˝ projection. In fact, we will prove that the solution to the analogous problem in the C∞ setting exists and is well behaved. We will then use the C∞ result to solve the classical problem.
Theorem 10.1. Suppose Ω is a bounded simply connected domain with C∞ smooth boundary and suppose ϕ is a function in C∞(bΩ). Let a ∈ Ω and let Sa(z) = S(z, a) and La(z) = L(z, a). Then, the function u = h+H, where h and H are holomorphic functions in A∞(Ω) given by
