Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions

In this paper, we consider the Cauchy problem of \begin{document}$ d $\end{document} -dimension Hartree type Dirac equation with nonlinearity \begin{document}$ c|x|^{-\gamma} * \langle \psi, \beta \psi\rangle $\end{document} , where \begin{document}$ c\in \mathbb R\setminus\{0\} $\end{document} , \begin{document}$ 0 < \gamma < d $\end{document} ( \begin{document}$ d = 2,3 $\end{document} ). Our aim is to show the local well-posedness in \begin{document}$ H^s $\end{document} for \begin{document}$ s > \frac{\gamma-1}2 $\end{document} with mass-supercritical cases( \begin{document}$ 1 < \gamma<d $\end{document} ) and mass-critical case( \begin{document}$ {\gamma} = 1 $\end{document} ) via bilinear estimates and angular decomposition for which we use the null structure of nonlinear term effectively. We also provide the flow of Dirac equations cannot be \begin{document}$ C^3 $\end{document} at the origin for \begin{document}$ H^s $\end{document} with \begin{document}$ s < \frac{\gamma-1}2 $\end{document} .