LAUSR

The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in \begin{document}$ L^2 $\end{document} -based Sobolev spaces \begin{document}$ H^s $\end{document} and \begin{document}$ H^l $\end{document} for the electromagnetic field \begin{document}$ \phi $\end{document} and the potential \begin{document}$ A $\end{document} , respectively, the minimal regularity assumptions are \begin{document}$ s > \frac{1}{2} $\end{document} and \begin{document}$ l > \frac{1}{4} $\end{document} , which leaves a gap of \begin{document}$ \frac{1}{2} $\end{document} and \begin{document}$ \frac{1}{4} $\end{document} to the critical regularity with respect to scaling \begin{document}$ s_c = l_c = 0 $\end{document} . This gap can be reduced for data in Fourier-Lebesgue spaces \begin{document}$ \widehat{H}^{s, r} $\end{document} and \begin{document}$ \widehat{H}^{l, r} $\end{document} to \begin{document}$ s> \frac{21}{16} $\end{document} and \begin{document}$ l > \frac{9}{8} $\end{document} for \begin{document}$ r $\end{document} close to \begin{document}$ 1 $\end{document} , whereas the critical exponents with respect to scaling fulfill \begin{document}$ s_c \to 1 $\end{document} , \begin{document}$ l_c \to 1 $\end{document} as \begin{document}$ r \to 1 $\end{document} . Here \begin{document}$ \|f\|_{\widehat{H}^{s, r}} : = \| \langle \xi \rangle^s \tilde{f}\|_{L^{r'}_{\tau \xi}} \, , \, 1 Thus the gap is reduced for \begin{document}$ \phi $\end{document} as well as \begin{document}$ A $\end{document} in both gauges.