  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
$L^2$-based Sobolev spaces $H^s$ and $H^l$ for the electromagnetic field $\phi$
and the potential $A$, respectively. The minimal regularity assumptions are $s
> \frac{1}{2}$ and $l > \frac{1}{4}$ , which leaves a gap of $\frac{1}{2}$ and
$\frac{1}{4}$ to the critical regularity with respect to scaling $s_c = l_c =0$
. This gap can be reduced for data in Fourier-Lebesgue spaces
$\widehat{H}^{s,r}$ and $\widehat{H}^{l,r}$ to $s> \frac{21}{16}$ and $l >
\frac{9}{8}$ for $r$ close to $1$ , whereas the critical exponents with respect
to scaling fulfill $s_c \to 1$ , $ l_c \to 1 $ as $r \to 1$ . Here
$\|f\|_{\widehat{H}^{s,r}} := \| \langle \xi \rangle^s
\tilde{f}\|_{L^{r'}_{\tau \xi}} \, , \, 1 < r \le 2 \, , \,
\frac{1}{r}+\frac{1}{r'} = 1 \, . $ Thus the gap is reduced for $\phi$ as well
as $A$ in both gauges.
