ABSTRACT
Suppose that Ω is a bounded simply connected domain with C∞ smooth boundary. Let a ∈ Ω be a fixed point. Given a real valued function u in C∞(bΩ), we may identify u as the boundary values of a real valued harmonic function U on Ω that is in C∞(Ω) (see Theorem 10.1). We will show that there is a real valued harmonic conjugate function V (meaning that U + iV is holomorphic on Ω) such that V is also in C∞(Ω). We can make V uniquely determined by specifying that V (a) = 0. Under these conditions, let v denote the restriction to the boundary of V . The function v is called the Hilbert transform of u and we write Hu = v. In this chapter, we will prove that the Hilbert transform is a well defined linear operator and we will prove the classical facts thatH maps C∞(bΩ) into itself and that H extends uniquely to be a bounded operator on L2(bΩ).
