This book studies some of the groundbreaking advances that have been made regarding analytic capacity and its relationship to rectifiability in the decade 1995-2005. The Cauchy transform plays a fundamental role in this area and is accordingly one of the main subjects covered. Another important topic, which may be of independent interest for many analysts, is the so-called non-homogeneous Calderon-Zygmund theory, the development of which has been largely motivated by the problems arising in connection with analytic capacity. The Painleve problem, which was first posed around 1900, consists in...

This book studies some of the groundbreaking advances that have been made regarding analytic capacity and its relationship to rectifiability. It provides a unified approach to the material and simplified proofs.