Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D

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 $\varphi$ and the potential $A$, respectively. The minimal regularity assumptions are $s

\frac{1}{2}$and$l > \frac{1}{4}$,whichleavesagapof$\frac{1}{2}$and$\frac{1}{4}$tothecriticalregularitywithrespecttoscaling$s_c = l_c =0$\mathrm{.}ThisgapcanbereducedfordatainFourier-Lebesguespaces$\widehat{H}^{s,r}$and$\widehat{H}^{l,r}$to$s> \frac{21}{16}$and$l > \frac{9}{8}$for$r$closeto$1$ , 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 , . $Thusthegapisreducedfor$\phi$ as well as $A$ in both gauges.