Dispersive estimates for massive Dirac operators in dimension two

We study the massive two dimensional Dirac operator with an electric potential. In particular, we show that the $t^{?1}$ decay rate holds in the $L^1\to L^{\infty }$ setting if the threshold energies are regular. We also show these bounds hold in the presence of s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of $t^{?1}{(\log ?t)}^{?2}$ is attained for large t, at the cost of logarithmic spatial weights. The free Dirac equation does not satisfy this bound due to the s-wave resonances at the threshold energies.