Dispersive Properties of the Dirac Equation : Strichartz estimates and the nonlinear problem

Ever since its invention in 1929 the Dirac equation has played a fundamental role in various areas of physics and mathematics. In this book we shall mainly deal with the PDE's approach, focusing on the well posedness for the cubic nonlinear problem for low regularity initial data. This kind of problems often requires a deep understanding of the algebraic structure of the operator; the investigation of the so called "critical" case turns out to be very hard for many dispersive equations (Schroedinger, wave and so on). The classical fixed-point strategy relies indeed on some space-time estimates (Strichartz estimates) which sometimes in the critical case are known to fail. This fact represents a huge obstacle, and some new approach is often needed. Part of the book will thus be devoted to the proof of some refined Strichartz estimates that will partially solve the critical problem.