  The purpose of this paper is to investigate several issues concerning the
Dirac equation from a time-frequency analysis perspective. More precisely, we
provide estimates in weighted modulation and Wiener amalgam spaces for the
solutions of the Dirac equation with rough potentials. We focus in particular
on bounded perturbations, arising as the Weyl quantization of suitable
time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth
functions, hence generalizing the results in a recent paper by Kato and
Naumkin. We then prove local well-posedness on the same function spaces for the
nonlinear Dirac equation with a general nonlinearity, including power-type
terms and the Thirring model. For this study we adopt the unifying framework of
vector-valued time-frequency analysis as developed by Wahlberg; most of the
preliminary results are stated under general assumptions and hence they may be
of independent interest.
