ABSTRACT
We have defined the Hardy space as a subspace of L2(bΩ). We will now identify the Hardy space with a space of holomorphic functions on Ω. If u ∈ L2(bΩ), then Cu has been defined to be the limit in L2(bΩ) of Cuj where uj is any sequence of functions in C∞(bΩ) converging to u in L2(bΩ). The functions Cuj are in A∞(Ω) and it is easy to see that they converge uniformly on compact subsets of Ω to a holomorphic function H . Although we have been thinking of the Cauchy transform C as an operator on L2(bΩ), let us agree to abuse our notation and also use the symbol C to represent the classical Cauchy integral,
(Cu)(z) = 1 2πi
∫ ζ∈bΩ
u(ζ)
ζ − z dζ
for z ∈ Ω. The holomorphic function H is given by H(z) = (Cu)(z). The purpose of this chapter is to show that this dual use of the symbol C is not an abuse. In fact, we will show that H has L2 boundary values given by Cu. Furthermore, there is a one-to-one correspondence between elements h of H2(bΩ) and holomorphic functions H on Ω arising as their Cauchy integrals. In this chapter, we will use lowercase letters to denote functions on the boundary in L2(bΩ) and we will let uppercase letters denote holomorphic functions on Ω. In particular, if h and H are used in the same paragraph, they will be related via the Cauchy integral formula H(z) = (Ch)(z). When this chapter is finished, we will be justified to use the same symbol for h and H .
