ABSTRACT
Today, Luigi Fantappiè’s name is widely known among complex analysts mainly in the context of the so-called Cauchy-Fantappiè Forms, even though few mathematicians know of the underlying connection. This chapter explains in simplest terms some of the basics of Fantappiè’s theory of analytic functionals, and traces the tenuous path leading to the machinery of Cauchy-Fantappiè formulas. It discusses some of the generalizations of the first formula of Cauchy-Fantappiè and some of the more recent applications, including a rather elementary solution of the Levi problem, a result which deals with the characterization of domains of holomorphy by local (pseudo-) convexity properties. The chapter also describes a new construction of Cauchy-Fantappiè forms with applications to estimates for solutions of the Cauchy-Riemann equations, which applies to pseudoconvex domains of finite type in two dimensions.
