On the propagation of the polarized wave front set

This thesis is devoted to the study of the propagation of singularities. First of all, we introduce fundamental tools from Microlocal analysis, that is pseudodifferential operators, Fourier integral operators and we recall some useful facts from symplectic geometry. The goal here is to develop a symbolic calculus that allows us to transform a problem involving a PDE into a problem of algebraic and geometric nature. Also, we present the notion of wave front set of a distribution and study its simplest properties. After that, following Hörmander’s work in the scalar case, we prove that the wave front set of a distribution is invariant under the bicharacteristic flow. Then, we deal with vector valued distributions, for which we introduce the polarized wave front set, as a refinement of the notion of wave front set. This definition, indeed, is able to detect the most singular components of a distribution. In the central part of the thesis, we study the propagation of the polarized wave front set for real principal type systems of pseudodifferential operators, for which the propagation is guaranteed. This is done by means of the so-called Dencker connection. Finally, we analyze the propagation of polarization in Maxwell’s system of equations and we prove that it is propagated along bicharacteristics without torsion.