\(L^2\) boundedness of the Cauchy integral and Menger curvature (Q2767870)

The author introduces the Menger curvature, which is defined as: Given three points in the plane, \(z_{1}\), \(z_{2}\) and \(z_{3}\), the Menger curvature associated to them is \(1/R\), where \(R\) is the radius of the circle passing through \(z_{1}\), \(z_{2}\) and \(z_{3}\). The basic properties of Menger curvature are then described. The Coifman-McIntosh-Meyer theorem on the Cauchy integral operator on a Lipschitz graph is then proved, using Menger curvature. Then a simple proof of the \(L^{2}\) boundedness of the first Calderón commutator is given, and the Cauchy integral is shown to be controlled by the first commutator. Finally, the various steps in the solution of Vitushkin's conjecture on analytic capacity are described, with special attention to the role played by Menger curvature.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00020].