Dispersive estimates for Dirac equations in Aharonov-Bohm magnetic fields: massless case

Abstract

In this paper we study the dispersive properties of a two dimensional massless Dirac equation perturbed by an Aharonov--Bohm magnetic field. Our main results will be a family of pointwise decay estimates and a full range family Strichartz estimates for the flow. The proof relies on the use of a relativistic Hankel transform, which allows for an explicit representation of the propagator in terms of the generalized eigenfunctions of the operator. These results represent the natural continuation of earlier research on evolution equations associated to operators with magnetic fields with strong singularities (see \cite{DF, FFFP, FZZ} where the Schr\"odinger and the wave equations were studied). Indeed, we recall the fact that the Aharonov--Bohm field represents a perturbation which is critical with respect to the scaling: this fact, as it is well known, makes the analysis particularly challenging.