A Real Variable Method for the Cauchy Transform, and Analytic Capacity [electronic resource] / by Takafumi Murai.

The calderón commutator (8 proofs of its boundedness) -- A real variable method for the cauchy transform on graphs -- Analytic capacities of cranks.

This research monograph studies the Cauchy transform on curves with the object of formulating a precise estimate of analytic capacity. The note is divided into three chapters. The first chapter is a review of the Calderón commutator. In the second chapter, a real variable method for the Cauchy transform is given using only the rising sun lemma. The final and principal chapter uses the method of the second chapter to compare analytic capacity with integral-geometric quantities. The prerequisites for reading this book are basic knowledge of singular integrals and function theory. It addresses specialists and graduate students in function theory and in fluid dynamics.