Boundedness of the Cauchy singular integral

Abstract

In this chapter our leading actor, the Cauchy singular integral operator S, enters the scene. A very deep theorem, which should actually be Theorem 1 of this book, says that S is bounded on L ^P(Γ, ω) (1 <p < ∞) if and only if Γ is a Carleson curve and ω is a Muckenhoupt weight in A _P (Γ). The proof of this theorem is difficult and goes beyond the scope of this book. We nevertheless decided to write down a proof, but this proof will only be given in Chapter 5. The purpose of this chapter is to provide the reader with some facts and results that should suffice to understand Chapters 6 to 10 without browsing in Chapter 5.